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)

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Normalized defining 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);
 

Invariants

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

\(\Q(\sqrt{-1}) \), 3.3.1849.1, 6.0.218803264.1, 7.7.6321363049.1, 14.0.654698590982350051753984.1, \(\Q(\zeta_{43})^+\)

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.14.14.38$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}$$[2]^{7}$
2.14.14.38$x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.38$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}$
43Data not computed