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Properties

Label 12.0.97738698162...8125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 7^{10}\cdot 11^{6}$
Root discriminant $56.13$
Ramified primes $5, 7, 11$
Class number $2512$ (GRH)
Class group $[2, 2, 628]$ (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![5417371, -1732597, 2225015, -537436, 463265, -59464, 52912, -3298, 3204, -94, 93, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 93*x^10 - 94*x^9 + 3204*x^8 - 3298*x^7 + 52912*x^6 - 59464*x^5 + 463265*x^4 - 537436*x^3 + 2225015*x^2 - 1732597*x + 5417371)
 
gp: K = bnfinit(x^12 - x^11 + 93*x^10 - 94*x^9 + 3204*x^8 - 3298*x^7 + 52912*x^6 - 59464*x^5 + 463265*x^4 - 537436*x^3 + 2225015*x^2 - 1732597*x + 5417371, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 93 x^{10} - 94 x^{9} + 3204 x^{8} - 3298 x^{7} + 52912 x^{6} - 59464 x^{5} + 463265 x^{4} - 537436 x^{3} + 2225015 x^{2} - 1732597 x + 5417371 \)

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:  \(977386981628298828125=5^{9}\cdot 7^{10}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.13$
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:  $5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(385=5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{385}(1,·)$, $\chi_{385}(307,·)$, $\chi_{385}(362,·)$, $\chi_{385}(331,·)$, $\chi_{385}(208,·)$, $\chi_{385}(254,·)$, $\chi_{385}(144,·)$, $\chi_{385}(309,·)$, $\chi_{385}(87,·)$, $\chi_{385}(153,·)$, $\chi_{385}(221,·)$, $\chi_{385}(318,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{370649} a^{10} - \frac{23894}{370649} a^{9} - \frac{17651}{370649} a^{8} - \frac{139036}{370649} a^{7} + \frac{74842}{370649} a^{6} - \frac{89719}{370649} a^{5} + \frac{165119}{370649} a^{4} + \frac{108991}{370649} a^{3} - \frac{129367}{370649} a^{2} + \frac{159828}{370649} a - \frac{44589}{370649}$, $\frac{1}{4243842870369648435596281} a^{11} - \frac{946103373349699283}{4243842870369648435596281} a^{10} - \frac{133818581127273460436354}{4243842870369648435596281} a^{9} + \frac{543404142346153957378418}{4243842870369648435596281} a^{8} - \frac{508293486824396154608461}{4243842870369648435596281} a^{7} - \frac{741796413009244306931070}{4243842870369648435596281} a^{6} + \frac{672060923046554189788278}{4243842870369648435596281} a^{5} - \frac{1193507682828028010522787}{4243842870369648435596281} a^{4} - \frac{2088313178844149381573335}{4243842870369648435596281} a^{3} + \frac{1119885607711837818402689}{4243842870369648435596281} a^{2} - \frac{550266005301090476024873}{4243842870369648435596281} a + \frac{194961007993651595}{783376820669961211}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{628}$, which has order $2512$ (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:  \( 104.882003477 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.741125.2, 6.6.300125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }$ R R R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$5$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$