# Properties

 Label 42.0.959...104.1 Degree $42$ Signature $[0, 21]$ Discriminant $-9.594\times 10^{77}$ Root discriminant $71.90$ Ramified primes $2, 43$ Class number $3756652$ (GRH) Class group $[2, 1878326]$ (GRH) Galois group $C_{42}$ (as 42T1)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 + 41*x^40 + 780*x^38 + 9139*x^36 + 73815*x^34 + 435897*x^32 + 1947792*x^30 + 6724520*x^28 + 18156204*x^26 + 38567100*x^24 + 64512240*x^22 + 84672315*x^20 + 86493225*x^18 + 67863915*x^16 + 40116600*x^14 + 17383860*x^12 + 5311735*x^10 + 1081575*x^8 + 134596*x^6 + 8855*x^4 + 231*x^2 + 1)

gp: K = bnfinit(x^42 + 41*x^40 + 780*x^38 + 9139*x^36 + 73815*x^34 + 435897*x^32 + 1947792*x^30 + 6724520*x^28 + 18156204*x^26 + 38567100*x^24 + 64512240*x^22 + 84672315*x^20 + 86493225*x^18 + 67863915*x^16 + 40116600*x^14 + 17383860*x^12 + 5311735*x^10 + 1081575*x^8 + 134596*x^6 + 8855*x^4 + 231*x^2 + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 231, 0, 8855, 0, 134596, 0, 1081575, 0, 5311735, 0, 17383860, 0, 40116600, 0, 67863915, 0, 86493225, 0, 84672315, 0, 64512240, 0, 38567100, 0, 18156204, 0, 6724520, 0, 1947792, 0, 435897, 0, 73815, 0, 9139, 0, 780, 0, 41, 0, 1]);

$$x^{42} + 41 x^{40} + 780 x^{38} + 9139 x^{36} + 73815 x^{34} + 435897 x^{32} + 1947792 x^{30} + 6724520 x^{28} + 18156204 x^{26} + 38567100 x^{24} + 64512240 x^{22} + 84672315 x^{20} + 86493225 x^{18} + 67863915 x^{16} + 40116600 x^{14} + 17383860 x^{12} + 5311735 x^{10} + 1081575 x^{8} + 134596 x^{6} + 8855 x^{4} + 231 x^{2} + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

 Degree: $42$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 21]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-95\!\cdots\!104$$$$\medspace = -\,2^{42}\cdot 43^{40}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $71.90$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $2, 43$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $42$ This field is Galois and abelian over $\Q$. Conductor: $$172=2^{2}\cdot 43$$ Dirichlet character group: $\lbrace$$\chi_{172}(1,·), \chi_{172}(17,·), \chi_{172}(133,·), \chi_{172}(135,·), \chi_{172}(9,·), \chi_{172}(11,·), \chi_{172}(13,·), \chi_{172}(15,·), \chi_{172}(145,·), \chi_{172}(21,·), \chi_{172}(23,·), \chi_{172}(25,·), \chi_{172}(31,·), \chi_{172}(35,·), \chi_{172}(165,·), \chi_{172}(167,·), \chi_{172}(41,·), \chi_{172}(47,·), \chi_{172}(49,·), \chi_{172}(53,·), \chi_{172}(57,·), \chi_{172}(59,·), \chi_{172}(67,·), \chi_{172}(79,·), \chi_{172}(81,·), \chi_{172}(83,·), \chi_{172}(139,·), \chi_{172}(87,·), \chi_{172}(143,·), \chi_{172}(95,·), \chi_{172}(97,·), \chi_{172}(99,·), \chi_{172}(101,·), \chi_{172}(103,·), \chi_{172}(107,·), \chi_{172}(109,·), \chi_{172}(111,·), \chi_{172}(117,·), \chi_{172}(169,·), \chi_{172}(153,·), \chi_{172}(121,·)$$\chi_{172}(127,·)$$\rbrace This is a CM field. ## Integral basis (with respect to field generator $$a$$) 1, a, a^{2}, a^{3}, a^{4}, a^{5}, a^{6}, a^{7}, a^{8}, a^{9}, a^{10}, a^{11}, a^{12}, a^{13}, a^{14}, a^{15}, a^{16}, a^{17}, a^{18}, a^{19}, a^{20}, a^{21}, a^{22}, a^{23}, a^{24}, a^{25}, a^{26}, a^{27}, a^{28}, a^{29}, a^{30}, a^{31}, a^{32}, a^{33}, a^{34}, a^{35}, a^{36}, a^{37}, a^{38}, a^{39}, a^{40}, a^{41} sage: K.integral_basis() gp: K.zk magma: IntegralBasis(K); ## Class group and class number C_{2}\times C_{1878326}, which has order 3756652 (assuming GRH) sage: K.class_group().invariants() gp: K.clgp magma: ClassGroup(K); ## Unit group sage: UK = K.unit_group() magma: UK, f := UnitGroup(K);  Rank: 20 sage: UK.rank() gp: K.fu magma: UnitRank(K); Torsion generator: $$-a^{41} - 40 a^{39} - 741 a^{37} - 8436 a^{35} - 66045 a^{33} - 376992 a^{31} - 1623160 a^{29} - 5379616 a^{27} - 13884156 a^{25} - 28048800 a^{23} - 44352165 a^{21} - 54627300 a^{19} - 51895935 a^{17} - 37442160 a^{15} - 20058300 a^{13} - 7726160 a^{11} - 2042975 a^{9} - 346104 a^{7} - 33649 a^{5} - 1540 a^{3} - 21 a$$ (order 4) sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) sage: UK.fundamental_units() gp: K.fu magma: [K!f(g): g in Generators(UK)]; Regulator: $$2748021948787771.5$$ (assuming GRH) sage: K.regulator() gp: K.reg magma: Regulator(K); ## Class number formula \displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{21}\cdot 2748021948787771.5 \cdot 3756652}{4\sqrt{959396304051793463814262846982490027578741814649477038563926538598268329263104}}\approx 0.152243719791663 (assuming GRH) ## Galois group C_{42} (as 42T1): sage: K.galois_group(type='pari') gp: polgalois(K.pol) magma: GaloisGroup(K);  A cyclic group of order 42 The 42 conjugacy class representatives for C_{42} Character table for C_{42} is not computed ## Intermediate fields Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities. ## Frobenius cycle types  p 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 Cycle type R 42 21^{2} {\href{/LocalNumberField/7.6.0.1}{6} }^{7} {\href{/LocalNumberField/11.14.0.1}{14} }^{3} 21^{2} 21^{2} 42 42 21^{2} 42 {\href{/LocalNumberField/37.3.0.1}{3} }^{14} {\href{/LocalNumberField/41.7.0.1}{7} }^{6} R {\href{/LocalNumberField/47.14.0.1}{14} }^{3} 21^{2} {\href{/LocalNumberField/59.14.0.1}{14} }^{3} In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents. sage: p = 7; # to obtain a list of [e_i,f_i] for the factorization of the ideal p\mathcal{O}_K: sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)] gp: p = 7; \\ to obtain a list of [e_i,f_i] for the factorization of the ideal p\mathcal{O}_K: gp: idealfactors = idealprimedec(K, p); \\ get the data gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]]) magma: p := 7; // to obtain a list of [e_i,f_i] for the factorization of the ideal p\mathcal{O}_K: magma: idealfactors := Factorization(p*Integers(K)); // get the data magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors]; ## Local algebras for ramified primes pLabelPolynomial e f c Galois group Slope content 22.14.14.38x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.38$x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$$2$$7$$14$$C_{14}$$[2]^{7} 2.14.14.38x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
43Data not computed