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Properties

Label 12.0.99194300950...5453.1
Degree $12$
Signature $[0, 6]$
Discriminant $7^{6}\cdot 13^{11}\cdot 19^{6}$
Root discriminant $121.07$
Ramified primes $7, 13, 19$
Class number $101732$ (GRH)
Class group $[2, 50866]$ (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![20571919369, -3782835772, 3782835772, -221515009, 221515009, -5677387, 5677387, -71215, 71215, -430, 430, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 430*x^10 - 430*x^9 + 71215*x^8 - 71215*x^7 + 5677387*x^6 - 5677387*x^5 + 221515009*x^4 - 221515009*x^3 + 3782835772*x^2 - 3782835772*x + 20571919369)
 
gp: K = bnfinit(x^12 - x^11 + 430*x^10 - 430*x^9 + 71215*x^8 - 71215*x^7 + 5677387*x^6 - 5677387*x^5 + 221515009*x^4 - 221515009*x^3 + 3782835772*x^2 - 3782835772*x + 20571919369, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 430 x^{10} - 430 x^{9} + 71215 x^{8} - 71215 x^{7} + 5677387 x^{6} - 5677387 x^{5} + 221515009 x^{4} - 221515009 x^{3} + 3782835772 x^{2} - 3782835772 x + 20571919369 \)

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:  \(9919430095046378756575453=7^{6}\cdot 13^{11}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $121.07$
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:  $7, 13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1729=7\cdot 13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1729}(1,·)$, $\chi_{1729}(930,·)$, $\chi_{1729}(132,·)$, $\chi_{1729}(134,·)$, $\chi_{1729}(265,·)$, $\chi_{1729}(398,·)$, $\chi_{1729}(400,·)$, $\chi_{1729}(531,·)$, $\chi_{1729}(1462,·)$, $\chi_{1729}(1065,·)$, $\chi_{1729}(932,·)$, $\chi_{1729}(666,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2199393769} a^{7} - \frac{923098515}{2199393769} a^{6} + \frac{231}{2199393769} a^{5} - \frac{223823143}{2199393769} a^{4} + \frac{15246}{2199393769} a^{3} + \frac{1017420151}{2199393769} a^{2} + \frac{251559}{2199393769} a + \frac{129768544}{2199393769}$, $\frac{1}{2199393769} a^{8} + \frac{264}{2199393769} a^{6} - \frac{329261771}{2199393769} a^{5} + \frac{21780}{2199393769} a^{4} + \frac{656652010}{2199393769} a^{3} + \frac{574992}{2199393769} a^{2} - \frac{324421360}{2199393769} a + \frac{2371842}{2199393769}$, $\frac{1}{2199393769} a^{9} - \frac{763962170}{2199393769} a^{6} - \frac{39204}{2199393769} a^{5} + \frac{362329999}{2199393769} a^{4} - \frac{3449952}{2199393769} a^{3} - \frac{597301406}{2199393769} a^{2} - \frac{64039734}{2199393769} a + \frac{931404688}{2199393769}$, $\frac{1}{2199393769} a^{10} - \frac{49005}{2199393769} a^{6} + \frac{886089749}{2199393769} a^{5} - \frac{5390550}{2199393769} a^{4} + \frac{979935559}{2199393769} a^{3} - \frac{160099335}{2199393769} a^{2} - \frac{536607502}{2199393769} a - \frac{704437074}{2199393769}$, $\frac{1}{2199393769} a^{11} - \frac{624990803}{2199393769} a^{6} + \frac{5929605}{2199393769} a^{5} + \frac{903538847}{2199393769} a^{4} + \frac{587030895}{2199393769} a^{3} + \frac{80542792}{2199393769} a^{2} + \frac{626242876}{2199393769} a + \frac{860112541}{2199393769}$

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

Class group and class number

$C_{2}\times C_{50866}$, which has order $101732$ (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:  \( 120.784031363 \) (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

\(\Q(\sqrt{13}) \), 3.3.169.1, 4.0.38862733.1, \(\Q(\zeta_{13})^+\)

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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
$7$7.12.6.2$x^{12} + 7203 x^{4} - 16807 x^{2} + 588245$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$13$13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$
$19$19.12.6.2$x^{12} - 2476099 x^{2} + 141137643$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$