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Properties

Label 12.8.97583054920...8301.1
Degree $12$
Signature $[8, 2]$
Discriminant $7^{8}\cdot 701^{5}$
Root discriminant $56.12$
Ramified primes $7, 701$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2^2.\SL(2,3)):C_2$ (as 12T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-48091, 259440, -335098, -57569, 64279, 6366, 1904, 452, -293, -39, -19, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 19*x^10 - 39*x^9 - 293*x^8 + 452*x^7 + 1904*x^6 + 6366*x^5 + 64279*x^4 - 57569*x^3 - 335098*x^2 + 259440*x - 48091)
 
gp: K = bnfinit(x^12 - 19*x^10 - 39*x^9 - 293*x^8 + 452*x^7 + 1904*x^6 + 6366*x^5 + 64279*x^4 - 57569*x^3 - 335098*x^2 + 259440*x - 48091, 1)
 

Normalized defining polynomial

\( x^{12} - 19 x^{10} - 39 x^{9} - 293 x^{8} + 452 x^{7} + 1904 x^{6} + 6366 x^{5} + 64279 x^{4} - 57569 x^{3} - 335098 x^{2} + 259440 x - 48091 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(975830549205637468301=7^{8}\cdot 701^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.12$
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:  $7, 701$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{131757366401253138451188792769} a^{11} + \frac{52112871755436356533314388809}{131757366401253138451188792769} a^{10} + \frac{96136399382323029673152074}{131757366401253138451188792769} a^{9} - \frac{7810087378478312780837602935}{131757366401253138451188792769} a^{8} - \frac{35531023691836888676539472861}{131757366401253138451188792769} a^{7} - \frac{57175450544873168579055355763}{131757366401253138451188792769} a^{6} + \frac{40739753710091900917997600450}{131757366401253138451188792769} a^{5} - \frac{3048414360506518590200085925}{10135182030865626034706830213} a^{4} - \frac{20507410088454899443613261930}{131757366401253138451188792769} a^{3} + \frac{41720257099141449530975227489}{131757366401253138451188792769} a^{2} - \frac{63522535255092084824790209860}{131757366401253138451188792769} a + \frac{34449218201711787411753753864}{131757366401253138451188792769}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 526156.374279 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2.\SL(2,3)):C_2$ (as 12T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 192
The 18 conjugacy class representatives for $(C_2^2.\SL(2,3)):C_2$
Character table for $(C_2^2.\SL(2,3)):C_2$

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.6.1683101.1

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

Sibling fields

Degree 12 siblings: 12.8.1985813112316901.1, 12.8.1985813112316901.2
Degree 24 siblings: data not computed
Arithmetically equvalently sibling: 12.8.975830549205637468301.2

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.12.0.1}{12} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
701Data not computed