# Properties

 Label 42.0.938...043.1 Degree $42$ Signature $[0, 21]$ Discriminant $-9.380\times 10^{66}$ Root discriminant $39.32$ Ramified prime $43$ Class number $211$ (GRH) Class group $[211]$ (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^41 + x^40 - x^39 + x^38 - x^37 + x^36 - x^35 + x^34 - x^33 + x^32 - x^31 + x^30 - x^29 + x^28 - x^27 + x^26 - x^25 + x^24 - x^23 + x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)

gp: K = bnfinit(x^42 - x^41 + x^40 - x^39 + x^38 - x^37 + x^36 - x^35 + x^34 - x^33 + x^32 - x^31 + x^30 - x^29 + x^28 - x^27 + x^26 - x^25 + x^24 - x^23 + x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1, 1)

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

$$x^{42} - x^{41} + x^{40} - x^{39} + x^{38} - x^{37} + x^{36} - x^{35} + x^{34} - x^{33} + x^{32} - x^{31} + x^{30} - x^{29} + x^{28} - x^{27} + x^{26} - x^{25} + x^{24} - x^{23} + x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$

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: $$-93\!\cdots\!043$$$$\medspace = -\,43^{41}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $39.32$ 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: $43$ 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: $$43$$ Dirichlet character group: $\lbrace$$\chi_{43}(1,·), \chi_{43}(2,·), \chi_{43}(3,·), \chi_{43}(4,·), \chi_{43}(5,·), \chi_{43}(6,·), \chi_{43}(7,·), \chi_{43}(8,·), \chi_{43}(9,·), \chi_{43}(10,·), \chi_{43}(11,·), \chi_{43}(12,·), \chi_{43}(13,·), \chi_{43}(14,·), \chi_{43}(15,·), \chi_{43}(16,·), \chi_{43}(17,·), \chi_{43}(18,·), \chi_{43}(19,·), \chi_{43}(20,·), \chi_{43}(21,·), \chi_{43}(22,·), \chi_{43}(23,·), \chi_{43}(24,·), \chi_{43}(25,·), \chi_{43}(26,·), \chi_{43}(27,·), \chi_{43}(28,·), \chi_{43}(29,·), \chi_{43}(30,·), \chi_{43}(31,·), \chi_{43}(32,·), \chi_{43}(33,·), \chi_{43}(34,·), \chi_{43}(35,·), \chi_{43}(36,·), \chi_{43}(37,·), \chi_{43}(38,·), \chi_{43}(39,·), \chi_{43}(40,·), \chi_{43}(41,·)$$\chi_{43}(42,·)$$\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

$C_{211}$, which has order $211$ (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 $86$) 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

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{21}\cdot 2748021948787771.5 \cdot 211}{86\sqrt{9380082945933081406113456619151991432292083579779389915131296484043}}\approx 0.127197359940775$ (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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{3}$ $42$ $42$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{7}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{6}$ $21^{2}$ $21^{2}$ $42$ $21^{2}$ $42$ $21^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{6}$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{6}$ $21^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{6}$

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
43Data not computed