↑

Properties

Label 12.4.20270094310...5881.1
Degree $12$
Signature $[4, 4]$
Discriminant $7^{8}\cdot 181^{6}$
Root discriminant $49.23$
Ramified primes $7, 181$
Class number $4$
Class group $[4]$
Galois group $C_4^2:C_3$ (as 12T31)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3375, -10800, -5700, -7117, -1015, 4501, -496, -65, -110, 18, -25, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 25*x^10 + 18*x^9 - 110*x^8 - 65*x^7 - 496*x^6 + 4501*x^5 - 1015*x^4 - 7117*x^3 - 5700*x^2 - 10800*x - 3375)
 
gp: K = bnfinit(x^12 - x^11 - 25*x^10 + 18*x^9 - 110*x^8 - 65*x^7 - 496*x^6 + 4501*x^5 - 1015*x^4 - 7117*x^3 - 5700*x^2 - 10800*x - 3375, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} - 25 x^{10} + 18 x^{9} - 110 x^{8} - 65 x^{7} - 496 x^{6} + 4501 x^{5} - 1015 x^{4} - 7117 x^{3} - 5700 x^{2} - 10800 x - 3375 \)

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:  \(202700943101784875881=7^{8}\cdot 181^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.23$
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:  $7, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{30} a^{7} + \frac{2}{15} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{4}{15} a - \frac{1}{2}$, $\frac{1}{30} a^{8} + \frac{2}{15} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{4}{15} a^{2} - \frac{1}{2} a$, $\frac{1}{210} a^{9} + \frac{1}{70} a^{8} - \frac{1}{105} a^{7} - \frac{1}{5} a^{6} + \frac{16}{35} a^{5} - \frac{13}{35} a^{4} - \frac{44}{105} a^{3} + \frac{29}{70} a^{2} - \frac{1}{70} a + \frac{3}{7}$, $\frac{1}{210} a^{10} + \frac{1}{70} a^{8} - \frac{1}{210} a^{7} + \frac{34}{105} a^{6} + \frac{34}{105} a^{5} + \frac{52}{105} a^{4} + \frac{33}{70} a^{3} + \frac{29}{105} a^{2} - \frac{41}{210} a + \frac{3}{14}$, $\frac{1}{241084090513772100} a^{11} - \frac{260520248468453}{120542045256886050} a^{10} - \frac{6972240971959}{9643363620550884} a^{9} + \frac{44292456018807}{26787121168196900} a^{8} - \frac{183541881011179}{48216818102754420} a^{7} - \frac{4944838956575879}{24108409051377210} a^{6} + \frac{13126003250656957}{120542045256886050} a^{5} - \frac{79644218727958289}{241084090513772100} a^{4} + \frac{8666508645752813}{24108409051377210} a^{3} - \frac{111022198747772407}{241084090513772100} a^{2} + \frac{1129350288260813}{5357424233639380} a - \frac{492401491197295}{1071484846727876}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( 234787.839792 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2:C_3$ (as 12T31):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 8 conjugacy class representatives for $C_4^2:C_3$
Character table for $C_4^2:C_3$

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.6.78659161.1

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

Sibling fields

Degree 12 sibling: data not computed
Degree 16 sibling: data not computed
Degree 24 sibling: 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.4.3.3$x^{4} + 362$$4$$1$$3$$C_4$$[\ ]_{4}$
181.4.3.3$x^{4} + 362$$4$$1$$3$$C_4$$[\ ]_{4}$