# Properties

 Label 42.0.118...207.1 Degree $42$ Signature $[0, 21]$ Discriminant $-1.182\times 10^{65}$ Root discriminant $35.43$ Ramified prime $7$ Class number $43$ (GRH) Class group $[43]$ (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 - x^35 + x^28 - x^21 + x^14 - x^7 + 1)

gp: K = bnfinit(x^42 - x^35 + x^28 - x^21 + x^14 - x^7 + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1]);

$$x^{42} - x^{35} + x^{28} - x^{21} + x^{14} - x^{7} + 1$$

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: $$-11\!\cdots\!207$$$$\medspace = -\,7^{77}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $35.43$ 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: $7$ 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: $$49=7^{2}$$ Dirichlet character group: $\lbrace$$\chi_{49}(1,·), \chi_{49}(2,·), \chi_{49}(3,·), \chi_{49}(4,·), \chi_{49}(5,·), \chi_{49}(6,·), \chi_{49}(8,·), \chi_{49}(9,·), \chi_{49}(10,·), \chi_{49}(11,·), \chi_{49}(12,·), \chi_{49}(13,·), \chi_{49}(15,·), \chi_{49}(16,·), \chi_{49}(17,·), \chi_{49}(18,·), \chi_{49}(19,·), \chi_{49}(20,·), \chi_{49}(22,·), \chi_{49}(23,·), \chi_{49}(24,·), \chi_{49}(25,·), \chi_{49}(26,·), \chi_{49}(27,·), \chi_{49}(29,·), \chi_{49}(30,·), \chi_{49}(31,·), \chi_{49}(32,·), \chi_{49}(33,·), \chi_{49}(34,·), \chi_{49}(36,·), \chi_{49}(37,·), \chi_{49}(38,·), \chi_{49}(39,·), \chi_{49}(40,·), \chi_{49}(41,·), \chi_{49}(43,·), \chi_{49}(44,·), \chi_{49}(45,·), \chi_{49}(46,·), \chi_{49}(47,·)$$\chi_{49}(48,·)$$\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

$C_{43}$, which has order $43$ (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: $$a$$ (order $98$) 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: $$1776855897760068.5$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{21}\cdot 1776855897760068.5 \cdot 43}{98\sqrt{118181386580595879976868414312001964434038548836769923458287039207}}\approx 0.131038008164406$ (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 $21^{2}$ $42$ $42$ R $21^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{3}$ $42$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{7}$ $21^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{7}$ $21^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{6}$ $42$ $21^{2}$ $42$

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
7Data not computed