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Properties

Label 12.0.11952055518...8125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 7^{8}\cdot 101^{6}$
Root discriminant $122.97$
Ramified primes $5, 7, 101$
Class number $262288$ (GRH)
Class group $[52, 5044]$ (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![39941506541, 1214690007, 4234486768, 87932303, 186438691, 2029774, 4325484, 10728, 55273, -116, 368, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 368*x^10 - 116*x^9 + 55273*x^8 + 10728*x^7 + 4325484*x^6 + 2029774*x^5 + 186438691*x^4 + 87932303*x^3 + 4234486768*x^2 + 1214690007*x + 39941506541)
 
gp: K = bnfinit(x^12 - x^11 + 368*x^10 - 116*x^9 + 55273*x^8 + 10728*x^7 + 4325484*x^6 + 2029774*x^5 + 186438691*x^4 + 87932303*x^3 + 4234486768*x^2 + 1214690007*x + 39941506541, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 368 x^{10} - 116 x^{9} + 55273 x^{8} + 10728 x^{7} + 4325484 x^{6} + 2029774 x^{5} + 186438691 x^{4} + 87932303 x^{3} + 4234486768 x^{2} + 1214690007 x + 39941506541 \)

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:  \(11952055518954678517578125=5^{9}\cdot 7^{8}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $122.97$
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:  $5, 7, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3535=5\cdot 7\cdot 101\)
Dirichlet character group:    $\lbrace$$\chi_{3535}(1,·)$, $\chi_{3535}(3334,·)$, $\chi_{3535}(807,·)$, $\chi_{3535}(809,·)$, $\chi_{3535}(1516,·)$, $\chi_{3535}(2829,·)$, $\chi_{3535}(302,·)$, $\chi_{3535}(2928,·)$, $\chi_{3535}(403,·)$, $\chi_{3535}(2423,·)$, $\chi_{3535}(1817,·)$, $\chi_{3535}(506,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{701} a^{9} + \frac{75}{701} a^{8} - \frac{61}{701} a^{7} - \frac{289}{701} a^{6} - \frac{11}{701} a^{5} + \frac{203}{701} a^{4} + \frac{210}{701} a^{3} + \frac{169}{701} a^{2} + \frac{208}{701} a + \frac{64}{701}$, $\frac{1}{701} a^{10} - \frac{78}{701} a^{8} + \frac{80}{701} a^{7} - \frac{67}{701} a^{6} + \frac{327}{701} a^{5} - \frac{294}{701} a^{4} - \frac{159}{701} a^{3} + \frac{151}{701} a^{2} - \frac{114}{701} a + \frac{107}{701}$, $\frac{1}{911076462885322170929450768392257203591} a^{11} - \frac{647061779792917868726747015744677915}{911076462885322170929450768392257203591} a^{10} - \frac{437526982157116162134703595387220633}{911076462885322170929450768392257203591} a^{9} - \frac{257959987384648467064992799456974947770}{911076462885322170929450768392257203591} a^{8} - \frac{194716607909650415418802211702995903949}{911076462885322170929450768392257203591} a^{7} + \frac{2198206657726986852083905576348878633}{31416429754666281756187957530767489779} a^{6} - \frac{420276633838320080872675790426747170094}{911076462885322170929450768392257203591} a^{5} - \frac{152125419004535641290379897366843028936}{911076462885322170929450768392257203591} a^{4} + \frac{187402264639288411323284326468720712375}{911076462885322170929450768392257203591} a^{3} + \frac{429732113870892731875891096251493892126}{911076462885322170929450768392257203591} a^{2} - \frac{43108054760825223852378388345553065764}{911076462885322170929450768392257203591} a + \frac{7457034367464304310196112804010722935}{31416429754666281756187957530767489779}$

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

Class group and class number

$C_{52}\times C_{5044}$, which has order $262288$ (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.1275125.2, 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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$101$101.6.3.1$x^{6} - 202 x^{4} + 10201 x^{2} - 124666421$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
101.6.3.1$x^{6} - 202 x^{4} + 10201 x^{2} - 124666421$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$