{
 "description": "Complete arithmetic specification of the fully stabilizing K3 x K3 flux of charge 22 (main text Section 3.4, Eqs. (40)-(43), Table 4; SM Section S6). Field K = Q(zeta_66), power basis 1, zeta, ..., zeta^19 of Z[zeta_66]; polynomials are coefficient lists c_0..c_n (value = sum c_i x^i); matrices are integer row lists acting on column vectors (column j = image of basis vector j). real_places lists k with sigma_k: zeta -> exp(2 pi i k/66), one per pair of complex embeddings. L = (Z[zeta_66], b), b(x,y) = Tr(eps_S delta^-1 x ybar), Gram B (20x20, symmetric Toeplitz with first row tau). Lambda = U + L with Gram d = [[0,1],[1,0]] + B in the basis (f1, f2, 1, zeta, ..., zeta^19). Flux: gtilde = gtilde_U + M_(1+zeta), g = g_U + M_(1+zeta^-1) (d-adjoint pair), S = g gtilde, Stilde = gtilde g; index convention g = N d, gtilde = N^T d, N_flux = (1/2) tr S. Kondo identification: G_BH and M_BH are the Gram matrix and distinguished generator of entry 0.66.0.1 of the K3Groups database (row convention: M_BH G_BH M_BH^T = G_BH); with A = M_BH^T and h = 1_U + M_zeta, the matrix P satisfies det P = 1, P^T G_BH P = d, A^5 P = P h (SM Eqs. (S6.15)-(S6.17)).",
 "field": {
  "n": 66,
  "degree": 20,
  "Phi66": [
   1,
   1,
   0,
   -1,
   -1,
   0,
   1,
   1,
   0,
   -1,
   -1,
   -1,
   0,
   1,
   1,
   0,
   -1,
   -1,
   0,
   1,
   1
  ],
  "abs_discriminant": "3^10 * 11^18",
  "real_places": [
   1,
   5,
   7,
   13,
   17,
   19,
   23,
   25,
   29,
   31
  ]
 },
 "delta": {
  "definition": "(zeta_3 - zeta_3^-1)(zeta_11 - zeta_11^-1)^9 with zeta_3 = zeta^22, zeta_11 = zeta^6; real; (delta) is the different of Z[zeta_66]",
  "power_basis": [
   -19,
   -16,
   -8,
   -17,
   186,
   -127,
   -145,
   150,
   -35,
   10,
   20,
   18,
   -2,
   -28,
   54,
   -83,
   -107,
   270,
   -85,
   -55
  ],
  "signs_of_delta_inverse_at_real_places": [
   -1,
   1,
   1,
   -1,
   -1,
   1,
   1,
   -1,
   -1,
   1
  ],
  "min_abs_delta_inverse_at_real_places": "0.0012363962945720638271510273849602304691095188193659990749041963257494820379433"
 },
 "eps_S": {
  "definition": "eps_S = -u_7 u_17 u_19 u_23 u_29, u_a = (zeta^a - zeta^-a)/(zeta - zeta^-1); a real unit of Z[zeta+zeta^-1]",
  "u_exponents": {
   "5": 0,
   "7": 1,
   "13": 0,
   "17": 1,
   "19": 1,
   "23": 1,
   "25": 0,
   "29": 1,
   "31": 0
  },
  "overall_sign": -1,
  "power_basis": [
   -2016,
   -4096,
   -4320,
   -2505,
   -566,
   -462,
   -2331,
   -4304,
   -4374,
   -2451,
   -358,
   1869,
   2227,
   582,
   -1170,
   -1087,
   1014,
   3186,
   3522,
   1806
  ],
  "condition": "sigma_k(eps delta^-1) > 0 for exactly one real place k; here k = 1",
  "signs_of_eps_delta_inverse_at_real_places": [
   1,
   -1,
   -1,
   -1,
   -1,
   -1,
   -1,
   -1,
   -1,
   -1
  ],
  "positive_place": 1,
  "min_abs_eps_delta_inverse_at_real_places": "8.9165415791729897621647662860158793141931851080186110784686644444344550230248e-6",
  "trace_eps_delta_inverse": 12726
 },
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   ]
  ],
  "det": 1,
  "signature": [
   2,
   18
  ],
  "even": true,
  "diagonal": 12726,
  "M_zeta": [
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    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
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   ]
  ],
  "M_zeta_note": "multiplication by zeta_66 in the power basis: integral isometry of L of order 66 with characteristic polynomial Phi66"
 },
 "Lambda": {
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  "signature": [
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  "rank": 22,
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  "N_max_abs_entry": 3846,
  "N_nonzero_entries": 398,
  "N_note": "N is basis dependent; these are its entries in the basis (f1, f2, 1, zeta, ..., zeta^19)",
  "tr_S": 44,
  "tr_S_split": {
   "U": 6,
   "L": 38
  },
  "det_S": 1,
  "N_flux": 22,
  "n_M2": 2,
  "budget": 24
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 "characteristic_polynomial": {
  "chi_S": [
   1,
   -86,
   2861,
   -48446,
   479766,
   -3015610,
   12725687,
   -37668104,
   80941466,
   -129649518,
   157932863,
   -148482242,
   108847544,
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   28311500,
   -10049472,
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   95209,
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   894,
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  "factorization": "(x^2 - 6x + 1) * m(x)^2",
  "m": [
   1,
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   390,
   -1443,
   2665,
   -2782,
   1742,
   -666,
   152,
   -19,
   1
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  "m_note": "m = minimal polynomial of 2 + zeta_66 + zeta_66^-1 = 2 + 2 cos(2 pi/66), irreducible of degree 10",
  "disc_m": "572981288913",
  "minimal_polynomial_of_S": "(x^2 - 6x + 1) * m(x) (squarefree, so S and Stilde are diagonalizable over R)"
 },
 "spectrum": {
  "note": "spec S = {3 +- 2 sqrt 2} on U and lambda_k = 2 + 2 cos(2 pi k/66), k in real_places, each of multiplicity 2 on the real plane P_k of L (x) R; b restricted to P_k is definite with the sign of sigma_k(eps_S delta^-1). Exactly three positive directions: the U-line of 3+2 sqrt 2 and the plane P_1; they span Sigma. Decimal strings are truncations.",
  "L_eigenplanes": [
   {
    "k": 1,
    "lambda_k": "3.9909438451461692094525105622598612315150537321589948701907084952313179149797",
    "sign_of_b_on_P_k": 1
   },
   {
    "k": 5,
    "lambda_k": "3.7776708973098469326231977859017091047356538884597351680803714025581420837761",
    "sign_of_b_on_P_k": -1
   },
   {
    "k": 7,
    "lambda_k": "3.5721061894855749395135921122944407320797337206519327681733739681061459371998",
    "sign_of_b_on_P_k": -1
   },
   {
    "k": 13,
    "lambda_k": "2.6541359266348432726834987403169048156145542533759802403146011965933342471658",
    "sign_of_b_on_P_k": -1
   },
   {
    "k": 17,
    "lambda_k": "1.9048361683525154051004255119370263302909311083813406292049288328051601300272",
    "sign_of_b_on_P_k": -1
   },
   {
    "k": 19,
    "lambda_k": "1.5284821289811455434989793593970248311472342777336692526198539984997664456769",
    "sign_of_b_on_P_k": -1
   },
   {
    "k": 23,
    "lambda_k": "0.83988618085760364160603773619938517553295608877007875466173816952556164691014",
    "sign_of_b_on_P_k": -1
   },
   {
    "k": 25,
    "lambda_k": "0.55253192378985967672028452647031977072785602542200683729115588350985293773055",
    "sign_of_b_on_P_k": -1
   },
   {
    "k": 29,
    "lambda_k": "0.14326413396785477959882255047281993033816355709609857499046045877584308359843",
    "sign_of_b_on_P_k": -1
   },
   {
    "k": 31,
    "lambda_k": "0.036142605474586599202651114750508078017863347950162904472807594394875572935489",
    "sign_of_b_on_P_k": -1
   }
  ],
  "min_gap_between_lambda_k": "0.10712152849326818039617143572231185232030020914593567051765286438096751066294",
  "n_positive_directions": 3,
  "Sigma": "line of 3+2 sqrt 2 in U (+) P_1",
  "Lambda_cap_Sigma_perp": "{0} (no lattice vector is orthogonal to Sigma or to Sigma-tilde; the smoothness clause of (FS) holds vacuously)"
 },
 "kondo": {
  "database": "S. Brandhorst and T. Hofmann, K3Groups (dataset), version 2.0, Zenodo doi:10.5281/zenodo.6544475 (CC-BY 4.0), entry results/purely_ns/order66.txt; S. Brandhorst, T. Hofmann, Forum Math. Sigma 11 (2023) e54, arXiv:2112.07715",
  "entry_id": "0.66.0.1",
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  ],
  "M_BH_convention": "acts on row vectors: M_BH * G_BH * M_BH^T = G_BH; A := M_BH^T acts on columns",
  "h": "1_U (+) M_zeta on the basis (f1, f2, 1, zeta, ..., zeta^19)",
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  ],
  "det_P": 1,
  "a": 5,
  "identities": [
   "P^T G_BH P = d",
   "A^5 P = P h",
   "A P = P h^53"
  ],
  "admissible_powers": [
   5,
   61
  ],
  "admissible_powers_note": "an isometry carrying h to A^a exists exactly for a in {5, 61}; composing P with complex conjugation on Z[zeta_66] (an isometry of L since eps_S delta^-1 is real) gives one for a = 61"
 },
 "not_decided": "Whether W = 0 at this vacuum (supersymmetry) is not decided by these data."
}