Properties

Label 42.42.981...729.1
Degree $42$
Signature $[42, 0]$
Discriminant $9.812\times 10^{76}$
Root discriminant $68.10$
Ramified primes $3, 43$
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 - x^41 - 42*x^40 + 42*x^39 + 818*x^38 - 818*x^37 - 9803*x^36 + 9803*x^35 + 80884*x^34 - 80884*x^33 - 487103*x^32 + 487103*x^31 + 2214673*x^30 - 2214673*x^29 - 7756167*x^28 + 7756167*x^27 + 21159269*x^26 - 21159269*x^25 - 45176143*x^24 + 45176143*x^23 + 75433697*x^22 - 75433697*x^21 - 97942948*x^20 + 97942948*x^19 + 97804877*x^18 - 97804877*x^17 - 73850908*x^16 + 73850908*x^15 + 41150012*x^14 - 41150012*x^13 - 16350448*x^12 + 16350448*x^11 + 4413607*x^10 - 4413607*x^9 - 753918*x^8 + 753918*x^7 + 72886*x^6 - 72886*x^5 - 3267*x^4 + 3267*x^3 + 44*x^2 - 44*x + 1)
 
gp: K = bnfinit(x^42 - x^41 - 42*x^40 + 42*x^39 + 818*x^38 - 818*x^37 - 9803*x^36 + 9803*x^35 + 80884*x^34 - 80884*x^33 - 487103*x^32 + 487103*x^31 + 2214673*x^30 - 2214673*x^29 - 7756167*x^28 + 7756167*x^27 + 21159269*x^26 - 21159269*x^25 - 45176143*x^24 + 45176143*x^23 + 75433697*x^22 - 75433697*x^21 - 97942948*x^20 + 97942948*x^19 + 97804877*x^18 - 97804877*x^17 - 73850908*x^16 + 73850908*x^15 + 41150012*x^14 - 41150012*x^13 - 16350448*x^12 + 16350448*x^11 + 4413607*x^10 - 4413607*x^9 - 753918*x^8 + 753918*x^7 + 72886*x^6 - 72886*x^5 - 3267*x^4 + 3267*x^3 + 44*x^2 - 44*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -44, 44, 3267, -3267, -72886, 72886, 753918, -753918, -4413607, 4413607, 16350448, -16350448, -41150012, 41150012, 73850908, -73850908, -97804877, 97804877, 97942948, -97942948, -75433697, 75433697, 45176143, -45176143, -21159269, 21159269, 7756167, -7756167, -2214673, 2214673, 487103, -487103, -80884, 80884, 9803, -9803, -818, 818, 42, -42, -1, 1]);
 

\( x^{42} - x^{41} - 42 x^{40} + 42 x^{39} + 818 x^{38} - 818 x^{37} - 9803 x^{36} + 9803 x^{35} + 80884 x^{34} - 80884 x^{33} - 487103 x^{32} + 487103 x^{31} + 2214673 x^{30} - 2214673 x^{29} - 7756167 x^{28} + 7756167 x^{27} + 21159269 x^{26} - 21159269 x^{25} - 45176143 x^{24} + 45176143 x^{23} + 75433697 x^{22} - 75433697 x^{21} - 97942948 x^{20} + 97942948 x^{19} + 97804877 x^{18} - 97804877 x^{17} - 73850908 x^{16} + 73850908 x^{15} + 41150012 x^{14} - 41150012 x^{13} - 16350448 x^{12} + 16350448 x^{11} + 4413607 x^{10} - 4413607 x^{9} - 753918 x^{8} + 753918 x^{7} + 72886 x^{6} - 72886 x^{5} - 3267 x^{4} + 3267 x^{3} + 44 x^{2} - 44 x + 1 \)

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[42, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(981\!\cdots\!729\)\(\medspace = 3^{21}\cdot 43^{41}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $68.10$
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:  $3, 43$
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:  \(129=3\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{129}(128,·)$, $\chi_{129}(1,·)$, $\chi_{129}(2,·)$, $\chi_{129}(4,·)$, $\chi_{129}(5,·)$, $\chi_{129}(8,·)$, $\chi_{129}(10,·)$, $\chi_{129}(13,·)$, $\chi_{129}(16,·)$, $\chi_{129}(20,·)$, $\chi_{129}(25,·)$, $\chi_{129}(26,·)$, $\chi_{129}(29,·)$, $\chi_{129}(31,·)$, $\chi_{129}(32,·)$, $\chi_{129}(40,·)$, $\chi_{129}(49,·)$, $\chi_{129}(50,·)$, $\chi_{129}(52,·)$, $\chi_{129}(58,·)$, $\chi_{129}(62,·)$, $\chi_{129}(64,·)$, $\chi_{129}(65,·)$, $\chi_{129}(67,·)$, $\chi_{129}(71,·)$, $\chi_{129}(77,·)$, $\chi_{129}(79,·)$, $\chi_{129}(80,·)$, $\chi_{129}(89,·)$, $\chi_{129}(97,·)$, $\chi_{129}(98,·)$, $\chi_{129}(100,·)$, $\chi_{129}(103,·)$, $\chi_{129}(104,·)$, $\chi_{129}(109,·)$, $\chi_{129}(113,·)$, $\chi_{129}(116,·)$, $\chi_{129}(119,·)$, $\chi_{129}(121,·)$, $\chi_{129}(124,·)$, $\chi_{129}(125,·)$$\chi_{129}(127,·)$$\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

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:  \( 16535537450237775000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{42}\cdot(2\pi)^{0}\cdot 16535537450237775000000000 \cdot 1}{2\sqrt{98118980687896783910098639727548084722605054105289047332129555342401833439729}}\approx 0.116083801936201$ (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

\(\Q(\sqrt{129}) \), 3.3.1849.1, 6.6.3969227961.1, 7.7.6321363049.1, 14.14.3757843639805369947326441.1, \(\Q(\zeta_{43})^+\)

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{6}$ R $21^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{7}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{3}$ $21^{2}$ $42$ $42$ $42$ $21^{2}$ $21^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{7}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{3}$ R ${\href{/LocalNumberField/47.14.0.1}{14} }^{3}$ $42$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{3}$

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
3Data not computed
43Data not computed