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Properties

Label 12.0.64856742023...3969.1
Degree $12$
Signature $[0, 6]$
Discriminant $13^{10}\cdot 19^{6}$
Root discriminant $36.95$
Ramified primes $13, 19$
Class number $117$
Class group $[117]$
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![73319, -23610, 44782, -14393, 13689, -4075, 2604, -686, 325, -68, 25, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 25*x^10 - 68*x^9 + 325*x^8 - 686*x^7 + 2604*x^6 - 4075*x^5 + 13689*x^4 - 14393*x^3 + 44782*x^2 - 23610*x + 73319)
 
gp: K = bnfinit(x^12 - 4*x^11 + 25*x^10 - 68*x^9 + 325*x^8 - 686*x^7 + 2604*x^6 - 4075*x^5 + 13689*x^4 - 14393*x^3 + 44782*x^2 - 23610*x + 73319, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 25 x^{10} - 68 x^{9} + 325 x^{8} - 686 x^{7} + 2604 x^{6} - 4075 x^{5} + 13689 x^{4} - 14393 x^{3} + 44782 x^{2} - 23610 x + 73319 \)

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:  \(6485674202367523969=13^{10}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.95$
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:  $13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(247=13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{247}(1,·)$, $\chi_{247}(134,·)$, $\chi_{247}(170,·)$, $\chi_{247}(75,·)$, $\chi_{247}(172,·)$, $\chi_{247}(77,·)$, $\chi_{247}(113,·)$, $\chi_{247}(246,·)$, $\chi_{247}(56,·)$, $\chi_{247}(153,·)$, $\chi_{247}(94,·)$, $\chi_{247}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{267262964521388687249} a^{11} + \frac{130701220091188902329}{267262964521388687249} a^{10} + \frac{9372938251824382847}{267262964521388687249} a^{9} + \frac{129367258292326811471}{267262964521388687249} a^{8} + \frac{96648670242224754516}{267262964521388687249} a^{7} + \frac{126817845002866729046}{267262964521388687249} a^{6} - \frac{13487050724512593795}{267262964521388687249} a^{5} + \frac{77222001124653202577}{267262964521388687249} a^{4} + \frac{13578261676513210740}{267262964521388687249} a^{3} + \frac{98669541064471744537}{267262964521388687249} a^{2} + \frac{111881866380088101744}{267262964521388687249} a + \frac{11763701975304220783}{267262964521388687249}$

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

Class group and class number

$C_{117}$, which has order $117$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 120.784031363 \)
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

\(\Q(\sqrt{-247}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\sqrt{13}, \sqrt{-19})\), 6.0.2546698687.1, 6.0.195899899.1, \(\Q(\zeta_{13})^+\)

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$13$13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$19$19.12.6.1$x^{12} + 41154 x^{6} - 2476099 x^{2} + 423412929$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$