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Properties

Label 12.0.28888480797...8125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 7^{8}\cdot 37^{6}$
Root discriminant $74.43$
Ramified primes $5, 7, 37$
Class number $13690$ (GRH)
Class group $[37, 370]$ (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![134106701, 7087687, 32886288, 1447263, 3493011, 100734, 205004, 1688, 6953, -36, 128, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 128*x^10 - 36*x^9 + 6953*x^8 + 1688*x^7 + 205004*x^6 + 100734*x^5 + 3493011*x^4 + 1447263*x^3 + 32886288*x^2 + 7087687*x + 134106701)
 
gp: K = bnfinit(x^12 - x^11 + 128*x^10 - 36*x^9 + 6953*x^8 + 1688*x^7 + 205004*x^6 + 100734*x^5 + 3493011*x^4 + 1447263*x^3 + 32886288*x^2 + 7087687*x + 134106701, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 128 x^{10} - 36 x^{9} + 6953 x^{8} + 1688 x^{7} + 205004 x^{6} + 100734 x^{5} + 3493011 x^{4} + 1447263 x^{3} + 32886288 x^{2} + 7087687 x + 134106701 \)

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:  \(28888480797518767578125=5^{9}\cdot 7^{8}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.43$
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:  $5, 7, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1295=5\cdot 7\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{1295}(704,·)$, $\chi_{1295}(1,·)$, $\chi_{1295}(998,·)$, $\chi_{1295}(519,·)$, $\chi_{1295}(1257,·)$, $\chi_{1295}(813,·)$, $\chi_{1295}(1072,·)$, $\chi_{1295}(149,·)$, $\chi_{1295}(186,·)$, $\chi_{1295}(443,·)$, $\chi_{1295}(926,·)$, $\chi_{1295}(702,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{109} a^{9} + \frac{27}{109} a^{8} - \frac{29}{109} a^{7} + \frac{35}{109} a^{6} - \frac{41}{109} a^{5} + \frac{37}{109} a^{4} - \frac{34}{109} a^{3} - \frac{2}{109} a^{2} - \frac{27}{109} a - \frac{30}{109}$, $\frac{1}{109} a^{10} + \frac{5}{109} a^{8} - \frac{54}{109} a^{7} - \frac{5}{109} a^{6} + \frac{54}{109} a^{5} - \frac{52}{109} a^{4} + \frac{44}{109} a^{3} + \frac{27}{109} a^{2} + \frac{45}{109} a + \frac{47}{109}$, $\frac{1}{22989983647052432303148594509879} a^{11} + \frac{34417094871172601513774954126}{22989983647052432303148594509879} a^{10} - \frac{7022146892953002903756175260}{22989983647052432303148594509879} a^{9} + \frac{9106202766678652590109184215974}{22989983647052432303148594509879} a^{8} + \frac{1527890191529816558170087604439}{22989983647052432303148594509879} a^{7} - \frac{11476949679396710985323893987807}{22989983647052432303148594509879} a^{6} + \frac{5450559081726941618207566448650}{22989983647052432303148594509879} a^{5} - \frac{7510409140734848772673553201156}{22989983647052432303148594509879} a^{4} - \frac{5212032905851160462071378372485}{22989983647052432303148594509879} a^{3} + \frac{9855914926945359087938568550386}{22989983647052432303148594509879} a^{2} - \frac{5418692396132287214668334608203}{22989983647052432303148594509879} a - \frac{318140993154872682583468438981}{792758056794911458729261879651}$

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

Class group and class number

$C_{37}\times C_{370}$, which has order $13690$ (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.171125.1, 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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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}$$[\ ]_{4}^{3}$
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$37$37.12.6.2$x^{12} - 69343957 x^{2} + 51314528180$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$