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Properties

Label 12.0.23345207816...7104.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{24}\cdot 7^{8}\cdot 17^{6}$
Root discriminant $60.35$
Ramified primes $2, 7, 17$
Class number $1512$ (GRH)
Class group $[3, 6, 84]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1308601, -390180, 599732, -149724, 121251, -24876, 13958, -2364, 1016, -136, 46, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 46*x^10 - 136*x^9 + 1016*x^8 - 2364*x^7 + 13958*x^6 - 24876*x^5 + 121251*x^4 - 149724*x^3 + 599732*x^2 - 390180*x + 1308601)
 
gp: K = bnfinit(x^12 - 4*x^11 + 46*x^10 - 136*x^9 + 1016*x^8 - 2364*x^7 + 13958*x^6 - 24876*x^5 + 121251*x^4 - 149724*x^3 + 599732*x^2 - 390180*x + 1308601, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 46 x^{10} - 136 x^{9} + 1016 x^{8} - 2364 x^{7} + 13958 x^{6} - 24876 x^{5} + 121251 x^{4} - 149724 x^{3} + 599732 x^{2} - 390180 x + 1308601 \)

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:  \(2334520781612309807104=2^{24}\cdot 7^{8}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.35$
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:  $2, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(952=2^{3}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{952}(1,·)$, $\chi_{952}(611,·)$, $\chi_{952}(613,·)$, $\chi_{952}(135,·)$, $\chi_{952}(137,·)$, $\chi_{952}(407,·)$, $\chi_{952}(205,·)$, $\chi_{952}(67,·)$, $\chi_{952}(681,·)$, $\chi_{952}(883,·)$, $\chi_{952}(477,·)$, $\chi_{952}(543,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{8} + \frac{1}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{9} a^{4} + \frac{1}{3} a^{3} - \frac{2}{9} a^{2} - \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{514827} a^{10} + \frac{25187}{514827} a^{9} - \frac{25172}{171609} a^{8} + \frac{42373}{514827} a^{7} + \frac{78815}{514827} a^{6} - \frac{220369}{514827} a^{5} - \frac{14864}{514827} a^{4} + \frac{54661}{514827} a^{3} + \frac{136540}{514827} a^{2} + \frac{172325}{514827} a + \frac{29663}{514827}$, $\frac{1}{2145945353313627} a^{11} + \frac{178057764}{238438372590403} a^{10} + \frac{27429423313027}{715315117771209} a^{9} - \frac{962028649705787}{2145945353313627} a^{8} + \frac{83552571928565}{715315117771209} a^{7} + \frac{714053790560999}{2145945353313627} a^{6} + \frac{81343080493760}{238438372590403} a^{5} + \frac{939865292930776}{2145945353313627} a^{4} + \frac{342857188348076}{2145945353313627} a^{3} - \frac{55803997658816}{2145945353313627} a^{2} - \frac{65160181259534}{238438372590403} a - \frac{4273958069497}{10074860813679}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{84}$, which has order $1512$ (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:  \( 279.1500271937239 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{-34}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-17}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{2}, \sqrt{-17})\), 6.0.6039609856.5, 6.6.1229312.1, 6.0.754951232.2

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.24.79$x^{12} - 4 x^{11} - 10 x^{10} + 16 x^{9} - 6 x^{8} + 16 x^{7} + 4 x^{6} - 8 x^{5} + 16 x^{4} + 16 x^{3} + 16 x^{2} + 8$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$17$17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$