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Properties

Label 12.0.23902002686...1328.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{33}\cdot 7^{8}\cdot 13^{6}$
Root discriminant $88.76$
Ramified primes $2, 7, 13$
Class number $11560$ (GRH)
Class group $[2, 34, 170]$ (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![75924577, -16329464, 23868696, -4102284, 3306578, -473884, 254842, -23140, 9167, -496, 154, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 154*x^10 - 496*x^9 + 9167*x^8 - 23140*x^7 + 254842*x^6 - 473884*x^5 + 3306578*x^4 - 4102284*x^3 + 23868696*x^2 - 16329464*x + 75924577)
 
gp: K = bnfinit(x^12 - 4*x^11 + 154*x^10 - 496*x^9 + 9167*x^8 - 23140*x^7 + 254842*x^6 - 473884*x^5 + 3306578*x^4 - 4102284*x^3 + 23868696*x^2 - 16329464*x + 75924577, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 154 x^{10} - 496 x^{9} + 9167 x^{8} - 23140 x^{7} + 254842 x^{6} - 473884 x^{5} + 3306578 x^{4} - 4102284 x^{3} + 23868696 x^{2} - 16329464 x + 75924577 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(239020026860167472611328=2^{33}\cdot 7^{8}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.76$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1456=2^{4}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1456}(1,·)$, $\chi_{1456}(571,·)$, $\chi_{1456}(417,·)$, $\chi_{1456}(1299,·)$, $\chi_{1456}(1353,·)$, $\chi_{1456}(779,·)$, $\chi_{1456}(883,·)$, $\chi_{1456}(625,·)$, $\chi_{1456}(51,·)$, $\chi_{1456}(729,·)$, $\chi_{1456}(1145,·)$, $\chi_{1456}(155,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{13} a^{6} - \frac{2}{13} a^{5} - \frac{3}{13} a^{4} + \frac{6}{13} a^{3} + \frac{2}{13} a^{2} - \frac{4}{13} a + \frac{1}{13}$, $\frac{1}{13} a^{7} + \frac{6}{13} a^{5} + \frac{1}{13} a^{3} + \frac{6}{13} a + \frac{2}{13}$, $\frac{1}{13} a^{8} - \frac{1}{13} a^{5} + \frac{6}{13} a^{4} + \frac{3}{13} a^{3} - \frac{6}{13} a^{2} - \frac{6}{13}$, $\frac{1}{13} a^{9} + \frac{4}{13} a^{5} + \frac{2}{13} a^{2} + \frac{3}{13} a + \frac{1}{13}$, $\frac{1}{80189154877} a^{10} - \frac{3021174960}{80189154877} a^{9} + \frac{411518004}{80189154877} a^{8} - \frac{1551664619}{80189154877} a^{7} + \frac{1526185695}{80189154877} a^{6} - \frac{4967043928}{80189154877} a^{5} + \frac{35244340489}{80189154877} a^{4} + \frac{20513217392}{80189154877} a^{3} + \frac{21234293167}{80189154877} a^{2} - \frac{4654069976}{80189154877} a - \frac{10967610390}{80189154877}$, $\frac{1}{20748490111699592132813} a^{11} - \frac{10560540528}{20748490111699592132813} a^{10} + \frac{592135868866584155042}{20748490111699592132813} a^{9} - \frac{42933077904382945010}{1596037700899968625601} a^{8} + \frac{324068564958798893863}{20748490111699592132813} a^{7} + \frac{64165570828549710004}{20748490111699592132813} a^{6} - \frac{961525474227098370572}{20748490111699592132813} a^{5} - \frac{5118745591923371179075}{20748490111699592132813} a^{4} - \frac{7736911281766860919470}{20748490111699592132813} a^{3} + \frac{3998822636783025233425}{20748490111699592132813} a^{2} + \frac{84038316985424138862}{506060734431697369093} a - \frac{6240536214172701391643}{20748490111699592132813}$

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

Class group and class number

$C_{2}\times C_{34}\times C_{170}$, which has order $11560$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 279.150027194 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), 4.0.346112.2, 6.6.1229312.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }$ R ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.12.33.376$x^{12} + 36 x^{10} + 42 x^{8} - 40 x^{6} + 40 x^{4} + 32 x^{2} - 56$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$