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Properties

Label 12.4.9979479338254336.4
Degree $12$
Signature $[4, 4]$
Discriminant $2^{24}\cdot 29^{6}$
Root discriminant $21.54$
Ramified primes $2, 29$
Class number $2$
Class group $[2]$
Galois group $C_2\times C_4^2:C_3:C_2$ (as 12T95)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -93, 543, -1003, 560, 144, -280, 101, -13, 7, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 5*x^11 + 7*x^10 - 13*x^9 + 101*x^8 - 280*x^7 + 144*x^6 + 560*x^5 - 1003*x^4 + 543*x^3 - 93*x^2 - x + 1)
 
gp: K = bnfinit(x^12 - 5*x^11 + 7*x^10 - 13*x^9 + 101*x^8 - 280*x^7 + 144*x^6 + 560*x^5 - 1003*x^4 + 543*x^3 - 93*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{12} - 5 x^{11} + 7 x^{10} - 13 x^{9} + 101 x^{8} - 280 x^{7} + 144 x^{6} + 560 x^{5} - 1003 x^{4} + 543 x^{3} - 93 x^{2} - x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9979479338254336=2^{24}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.54$
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:  $2, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{379412615} a^{11} - \frac{4995305}{75882523} a^{10} + \frac{21088588}{379412615} a^{9} - \frac{128945069}{379412615} a^{8} + \frac{34806097}{379412615} a^{7} - \frac{104887176}{379412615} a^{6} - \frac{50711219}{379412615} a^{5} - \frac{18245048}{379412615} a^{4} + \frac{37473993}{75882523} a^{3} + \frac{44991643}{379412615} a^{2} - \frac{25195874}{379412615} a - \frac{24477308}{75882523}$

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

Class group and class number

$C_{2}$, which has order $2$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1800.10972161 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2:C_3:C_2$ (as 12T95):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 192
The 20 conjugacy class representatives for $C_2\times C_4^2:C_3:C_2$
Character table for $C_2\times C_4^2:C_3:C_2$

Intermediate fields

3.1.116.1, 6.2.215296.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.24.73$x^{8} + 8 x^{7} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 6$$8$$1$$24$$C_4\wr C_2$$[2, 3, 4]^{4}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$