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Properties

Label 42.42.412...472.1
Degree $42$
Signature $[42, 0]$
Discriminant $4.125\times 10^{79}$
Root discriminant $78.63$
Ramified primes $2, 43$
Class number not computed
Class group not computed
Galois group $C_{42}$ (as 42T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 - 43*x^40 + 860*x^38 - 10621*x^36 + 90687*x^34 - 567987*x^32 + 2701776*x^30 - 9970840*x^28 + 28915436*x^26 - 66335412*x^24 + 120609840*x^22 - 173376645*x^20 + 195747825*x^18 - 171655785*x^16 + 115000920*x^14 - 57500460*x^12 + 20764055*x^10 - 5167525*x^8 + 826804*x^6 - 76153*x^4 + 3311*x^2 - 43)
 
gp: K = bnfinit(x^42 - 43*x^40 + 860*x^38 - 10621*x^36 + 90687*x^34 - 567987*x^32 + 2701776*x^30 - 9970840*x^28 + 28915436*x^26 - 66335412*x^24 + 120609840*x^22 - 173376645*x^20 + 195747825*x^18 - 171655785*x^16 + 115000920*x^14 - 57500460*x^12 + 20764055*x^10 - 5167525*x^8 + 826804*x^6 - 76153*x^4 + 3311*x^2 - 43, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-43, 0, 3311, 0, -76153, 0, 826804, 0, -5167525, 0, 20764055, 0, -57500460, 0, 115000920, 0, -171655785, 0, 195747825, 0, -173376645, 0, 120609840, 0, -66335412, 0, 28915436, 0, -9970840, 0, 2701776, 0, -567987, 0, 90687, 0, -10621, 0, 860, 0, -43, 0, 1]);
 

\( x^{42} - 43 x^{40} + 860 x^{38} - 10621 x^{36} + 90687 x^{34} - 567987 x^{32} + 2701776 x^{30} - 9970840 x^{28} + 28915436 x^{26} - 66335412 x^{24} + 120609840 x^{22} - 173376645 x^{20} + 195747825 x^{18} - 171655785 x^{16} + 115000920 x^{14} - 57500460 x^{12} + 20764055 x^{10} - 5167525 x^{8} + 826804 x^{6} - 76153 x^{4} + 3311 x^{2} - 43 \)

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:  \(412\!\cdots\!472\)\(\medspace = 2^{42}\cdot 43^{41}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $78.63$
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$$\chi_{172}(1,·)$, $\chi_{172}(3,·)$, $\chi_{172}(133,·)$, $\chi_{172}(7,·)$, $\chi_{172}(9,·)$, $\chi_{172}(13,·)$, $\chi_{172}(115,·)$, $\chi_{172}(17,·)$, $\chi_{172}(19,·)$, $\chi_{172}(21,·)$, $\chi_{172}(151,·)$, $\chi_{172}(25,·)$, $\chi_{172}(155,·)$, $\chi_{172}(159,·)$, $\chi_{172}(27,·)$, $\chi_{172}(165,·)$, $\chi_{172}(39,·)$, $\chi_{172}(41,·)$, $\chi_{172}(171,·)$, $\chi_{172}(49,·)$, $\chi_{172}(51,·)$, $\chi_{172}(53,·)$, $\chi_{172}(55,·)$, $\chi_{172}(57,·)$, $\chi_{172}(63,·)$, $\chi_{172}(71,·)$, $\chi_{172}(75,·)$, $\chi_{172}(81,·)$, $\chi_{172}(163,·)$, $\chi_{172}(169,·)$, $\chi_{172}(91,·)$, $\chi_{172}(97,·)$, $\chi_{172}(131,·)$, $\chi_{172}(101,·)$, $\chi_{172}(145,·)$, $\chi_{172}(109,·)$, $\chi_{172}(147,·)$, $\chi_{172}(117,·)$, $\chi_{172}(119,·)$, $\chi_{172}(153,·)$, $\chi_{172}(121,·)$, $\chi_{172}(123,·)$$\rbrace$
This is not 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

not computed

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:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

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

\(\Q(\sqrt{43}) \), 3.3.1849.1, 6.6.9408540352.1, 7.7.6321363049.1, 14.14.28152039412241052225421312.1, \(\Q(\zeta_{43})^+\)

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 $21^{2}$ $42$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{14}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{3}$ $21^{2}$ $21^{2}$ $21^{2}$ $42$ $42$ $42$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{6}$ 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$2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
43Data not computed