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Properties

Label 12.0.53483214216...3125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 7^{8}\cdot 41^{6}$
Root discriminant $78.35$
Ramified primes $5, 7, 41$
Class number $13952$ (GRH)
Class group $[2, 2, 2, 2, 2, 436]$ (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![236593391, 12100257, 53594968, 2217128, 5232016, 136924, 280434, 2028, 8623, -41, 143, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 143*x^10 - 41*x^9 + 8623*x^8 + 2028*x^7 + 280434*x^6 + 136924*x^5 + 5232016*x^4 + 2217128*x^3 + 53594968*x^2 + 12100257*x + 236593391)
 
gp: K = bnfinit(x^12 - x^11 + 143*x^10 - 41*x^9 + 8623*x^8 + 2028*x^7 + 280434*x^6 + 136924*x^5 + 5232016*x^4 + 2217128*x^3 + 53594968*x^2 + 12100257*x + 236593391, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 143 x^{10} - 41 x^{9} + 8623 x^{8} + 2028 x^{7} + 280434 x^{6} + 136924 x^{5} + 5232016 x^{4} + 2217128 x^{3} + 53594968 x^{2} + 12100257 x + 236593391 \)

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:  \(53483214216056720703125=5^{9}\cdot 7^{8}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.35$
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:  $5, 7, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1435=5\cdot 7\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{1435}(1,·)$, $\chi_{1435}(1026,·)$, $\chi_{1435}(739,·)$, $\chi_{1435}(737,·)$, $\chi_{1435}(1352,·)$, $\chi_{1435}(778,·)$, $\chi_{1435}(942,·)$, $\chi_{1435}(368,·)$, $\chi_{1435}(163,·)$, $\chi_{1435}(821,·)$, $\chi_{1435}(534,·)$, $\chi_{1435}(1149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{131} a^{9} + \frac{30}{131} a^{8} - \frac{31}{131} a^{7} - \frac{31}{131} a^{6} - \frac{21}{131} a^{5} + \frac{56}{131} a^{4} - \frac{51}{131} a^{3} + \frac{32}{131} a^{2} - \frac{42}{131} a - \frac{6}{131}$, $\frac{1}{131} a^{10} - \frac{14}{131} a^{8} - \frac{18}{131} a^{7} - \frac{8}{131} a^{6} + \frac{31}{131} a^{5} - \frac{28}{131} a^{4} - \frac{10}{131} a^{3} + \frac{46}{131} a^{2} - \frac{56}{131} a + \frac{49}{131}$, $\frac{1}{127792074940964081489684316424421} a^{11} - \frac{214584703044963069896769205499}{127792074940964081489684316424421} a^{10} - \frac{299542511127706209553069808354}{127792074940964081489684316424421} a^{9} + \frac{19208061633403008045535982757779}{127792074940964081489684316424421} a^{8} - \frac{46023543781733271239575654660917}{127792074940964081489684316424421} a^{7} + \frac{30072602105700121081420638503895}{127792074940964081489684316424421} a^{6} + \frac{15365392398087512689865026229743}{127792074940964081489684316424421} a^{5} + \frac{20626597036007386822248821632508}{127792074940964081489684316424421} a^{4} - \frac{40626672019148579955304827735707}{127792074940964081489684316424421} a^{3} + \frac{21814061365780918567079427082490}{127792074940964081489684316424421} a^{2} + \frac{42465060521591389537938109487846}{127792074940964081489684316424421} a + \frac{2542750945359639029009387878595}{127792074940964081489684316424421}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{436}$, which has order $13952$ (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.210125.1, 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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$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}$
$41$41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$