Properties

Label 42.42.155...608.1
Degree $42$
Signature $[42, 0]$
Discriminant $1.557\times 10^{83}$
Root discriminant $95.67$
Ramified primes $2, 7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{42}$ (as 42T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 - 84*x^40 + 3276*x^38 - 78736*x^36 + 1305360*x^34 - 15833664*x^32 + 145435136*x^30 - 1032886144*x^28 + 5741625344*x^26 - 25131545600*x^24 + 86701812736*x^22 - 234915702784*x^20 + 495818960896*x^18 - 804378279936*x^16 + 983365402624*x^14 - 880319397888*x^12 + 553414557696*x^10 - 229269569536*x^8 + 56479711232*x^6 - 6910640128*x^4 + 308281344*x^2 - 2097152)
 
gp: K = bnfinit(x^42 - 84*x^40 + 3276*x^38 - 78736*x^36 + 1305360*x^34 - 15833664*x^32 + 145435136*x^30 - 1032886144*x^28 + 5741625344*x^26 - 25131545600*x^24 + 86701812736*x^22 - 234915702784*x^20 + 495818960896*x^18 - 804378279936*x^16 + 983365402624*x^14 - 880319397888*x^12 + 553414557696*x^10 - 229269569536*x^8 + 56479711232*x^6 - 6910640128*x^4 + 308281344*x^2 - 2097152, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2097152, 0, 308281344, 0, -6910640128, 0, 56479711232, 0, -229269569536, 0, 553414557696, 0, -880319397888, 0, 983365402624, 0, -804378279936, 0, 495818960896, 0, -234915702784, 0, 86701812736, 0, -25131545600, 0, 5741625344, 0, -1032886144, 0, 145435136, 0, -15833664, 0, 1305360, 0, -78736, 0, 3276, 0, -84, 0, 1]);
 

\( x^{42} - 84 x^{40} + 3276 x^{38} - 78736 x^{36} + 1305360 x^{34} - 15833664 x^{32} + 145435136 x^{30} - 1032886144 x^{28} + 5741625344 x^{26} - 25131545600 x^{24} + 86701812736 x^{22} - 234915702784 x^{20} + 495818960896 x^{18} - 804378279936 x^{16} + 983365402624 x^{14} - 880319397888 x^{12} + 553414557696 x^{10} - 229269569536 x^{8} + 56479711232 x^{6} - 6910640128 x^{4} + 308281344 x^{2} - 2097152 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[42, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(155\!\cdots\!608\)\(\medspace = 2^{63}\cdot 7^{76}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $95.67$
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:  $2, 7$
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:  \(392=2^{3}\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{392}(1,·)$, $\chi_{392}(261,·)$, $\chi_{392}(65,·)$, $\chi_{392}(9,·)$, $\chi_{392}(141,·)$, $\chi_{392}(365,·)$, $\chi_{392}(277,·)$, $\chi_{392}(25,·)$, $\chi_{392}(281,·)$, $\chi_{392}(29,·)$, $\chi_{392}(389,·)$, $\chi_{392}(289,·)$, $\chi_{392}(37,·)$, $\chi_{392}(305,·)$, $\chi_{392}(169,·)$, $\chi_{392}(93,·)$, $\chi_{392}(177,·)$, $\chi_{392}(53,·)$, $\chi_{392}(137,·)$, $\chi_{392}(57,·)$, $\chi_{392}(317,·)$, $\chi_{392}(309,·)$, $\chi_{392}(193,·)$, $\chi_{392}(197,·)$, $\chi_{392}(333,·)$, $\chi_{392}(205,·)$, $\chi_{392}(337,·)$, $\chi_{392}(85,·)$, $\chi_{392}(121,·)$, $\chi_{392}(345,·)$, $\chi_{392}(221,·)$, $\chi_{392}(165,·)$, $\chi_{392}(225,·)$, $\chi_{392}(81,·)$, $\chi_{392}(361,·)$, $\chi_{392}(109,·)$, $\chi_{392}(113,·)$, $\chi_{392}(373,·)$, $\chi_{392}(233,·)$, $\chi_{392}(249,·)$, $\chi_{392}(253,·)$$\chi_{392}(149,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{4096} a^{24}$, $\frac{1}{4096} a^{25}$, $\frac{1}{8192} a^{26}$, $\frac{1}{8192} a^{27}$, $\frac{1}{16384} a^{28}$, $\frac{1}{16384} a^{29}$, $\frac{1}{32768} a^{30}$, $\frac{1}{32768} a^{31}$, $\frac{1}{65536} a^{32}$, $\frac{1}{65536} a^{33}$, $\frac{1}{131072} a^{34}$, $\frac{1}{131072} a^{35}$, $\frac{1}{262144} a^{36}$, $\frac{1}{262144} a^{37}$, $\frac{1}{524288} a^{38}$, $\frac{1}{524288} a^{39}$, $\frac{1}{1048576} a^{40}$, $\frac{1}{1048576} a^{41}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

Trivial group, which has order $1$ (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:  $41$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 31575205777918616000000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{42}\cdot(2\pi)^{0}\cdot 31575205777918616000000000000 \cdot 1}{2\sqrt{155718699466313184257207094263668545441599708733396657696588937331033553383727300608}}\approx 0.175956626859625$ (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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), 6.6.1229312.1, 7.7.13841287201.1, 14.14.401774962552217617093885952.1, \(\Q(\zeta_{49})^+\)

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 $42$ $42$ R $42$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{3}$ $21^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{7}$ $21^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{14}$ $42$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{6}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{3}$ $21^{2}$ $42$ $42$

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
2Data not computed
7Data not computed