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Properties

Label 12.0.31599259021...0000.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{16}\cdot 5^{6}\cdot 13^{11}$
Root discriminant $287.28$
Ramified primes $2, 3, 5, 13$
Class number $2601612$ (GRH)
Class group $[3, 6, 144534]$ (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![17866171479, 4707053910, 1693748589, -15701114, 37302525, -1148862, 2844842, -6162, 45123, -52, 312, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 312*x^10 - 52*x^9 + 45123*x^8 - 6162*x^7 + 2844842*x^6 - 1148862*x^5 + 37302525*x^4 - 15701114*x^3 + 1693748589*x^2 + 4707053910*x + 17866171479)
 
gp: K = bnfinit(x^12 + 312*x^10 - 52*x^9 + 45123*x^8 - 6162*x^7 + 2844842*x^6 - 1148862*x^5 + 37302525*x^4 - 15701114*x^3 + 1693748589*x^2 + 4707053910*x + 17866171479, 1)
 

Normalized defining polynomial

\( x^{12} + 312 x^{10} - 52 x^{9} + 45123 x^{8} - 6162 x^{7} + 2844842 x^{6} - 1148862 x^{5} + 37302525 x^{4} - 15701114 x^{3} + 1693748589 x^{2} + 4707053910 x + 17866171479 \)

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:  \(315992590210501847764992000000=2^{18}\cdot 3^{16}\cdot 5^{6}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $287.28$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4680=2^{3}\cdot 3^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{4680}(1921,·)$, $\chi_{4680}(3109,·)$, $\chi_{4680}(1,·)$, $\chi_{4680}(3721,·)$, $\chi_{4680}(109,·)$, $\chi_{4680}(2401,·)$, $\chi_{4680}(3469,·)$, $\chi_{4680}(1681,·)$, $\chi_{4680}(709,·)$, $\chi_{4680}(4309,·)$, $\chi_{4680}(2521,·)$, $\chi_{4680}(3349,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{13} a^{6}$, $\frac{1}{13} a^{7}$, $\frac{1}{39} a^{8} - \frac{1}{39} a^{7} + \frac{1}{39} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{39} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{47541} a^{10} - \frac{32}{47541} a^{9} - \frac{2}{689} a^{8} - \frac{295}{15847} a^{7} - \frac{359}{15847} a^{6} + \frac{511}{1219} a^{5} - \frac{1792}{3657} a^{4} - \frac{1390}{3657} a^{3} + \frac{110}{1219} a^{2} + \frac{534}{1219} a + \frac{103}{1219}$, $\frac{1}{41456366853239948093778709589467778530461} a^{11} + \frac{22580143057616757740232460804700530}{3188951296403072930290669968420598348497} a^{10} + \frac{408074067331654014413584023515591562370}{41456366853239948093778709589467778530461} a^{9} - \frac{54459096196458870135656815374382295327}{13818788951079982697926236529822592843487} a^{8} + \frac{5005162837705336064585822427968540479}{13818788951079982697926236529822592843487} a^{7} - \frac{255493715287176013532950082553399318509}{13818788951079982697926236529822592843487} a^{6} - \frac{552129810504670456157947299692883118312}{3188951296403072930290669968420598348497} a^{5} - \frac{1100107540675814872847870349700309185175}{3188951296403072930290669968420598348497} a^{4} + \frac{36722237062821319207323284981962403452}{138650056365350996969159563844373841239} a^{3} - \frac{155700377566824048236626130105820480206}{354327921822563658921185552046733149833} a^{2} + \frac{63194705146537572136339194748859176901}{1062983765467690976763556656140199449499} a - \frac{481936916475540565193018251695240468277}{1062983765467690976763556656140199449499}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{144534}$, which has order $2601612$ (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:  \( 9116.238746847283 \) (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

\(\Q(\sqrt{13}) \), 3.3.13689.2, 4.0.3515200.1, 6.6.2436053373.2

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$2.12.18.28$x^{12} - 52 x^{10} + 1100 x^{8} - 12000 x^{6} - 61072 x^{4} + 62144 x^{2} - 62144$$2$$6$$18$$C_{12}$$[3]^{6}$
$3$3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
$5$5.12.6.2$x^{12} - 3125 x^{2} + 31250$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$13$13.12.11.2$x^{12} - 52$$12$$1$$11$$C_{12}$$[\ ]_{12}$