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Properties

Label 12.0.16384000000000000.7
Degree $12$
Signature $[0, 6]$
Discriminant $2^{26}\cdot 5^{12}$
Root discriminant $22.45$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_5$ (as 12T74)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![196, 112, 32, 0, -140, 16, 32, -8, 25, -20, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 8*x^10 - 20*x^9 + 25*x^8 - 8*x^7 + 32*x^6 + 16*x^5 - 140*x^4 + 32*x^2 + 112*x + 196)
 
gp: K = bnfinit(x^12 - 4*x^11 + 8*x^10 - 20*x^9 + 25*x^8 - 8*x^7 + 32*x^6 + 16*x^5 - 140*x^4 + 32*x^2 + 112*x + 196, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 8 x^{10} - 20 x^{9} + 25 x^{8} - 8 x^{7} + 32 x^{6} + 16 x^{5} - 140 x^{4} + 32 x^{2} + 112 x + 196 \)

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:  \(16384000000000000=2^{26}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.45$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{4} + \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{20} a^{8} + \frac{3}{10} a^{6} + \frac{2}{5} a^{5} - \frac{9}{20} a^{4} - \frac{1}{5} a^{3} - \frac{1}{10} a^{2} + \frac{2}{5} a + \frac{1}{10}$, $\frac{1}{20} a^{9} - \frac{1}{10} a^{7} + \frac{1}{5} a^{6} - \frac{9}{20} a^{5} - \frac{2}{5} a^{4} - \frac{1}{10} a^{3} - \frac{3}{10} a - \frac{2}{5}$, $\frac{1}{40} a^{10} - \frac{1}{10} a^{7} - \frac{1}{40} a^{6} - \frac{3}{10} a^{5} - \frac{1}{10} a^{4} - \frac{1}{5} a^{3} - \frac{9}{20} a^{2} - \frac{1}{10}$, $\frac{1}{4461800} a^{11} - \frac{11537}{1115450} a^{10} + \frac{5927}{446180} a^{9} - \frac{9459}{446180} a^{8} - \frac{88209}{892360} a^{7} + \frac{216123}{1115450} a^{6} - \frac{532593}{2230900} a^{5} - \frac{80763}{446180} a^{4} + \frac{30027}{63740} a^{3} - \frac{8819}{31870} a^{2} - \frac{71641}{557725} a - \frac{64047}{159350}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( -\frac{6603}{1115450} a^{11} + \frac{60153}{2230900} a^{10} - \frac{1199}{22309} a^{9} + \frac{14886}{111545} a^{8} - \frac{43963}{223090} a^{7} + \frac{260943}{2230900} a^{6} - \frac{190726}{557725} a^{5} + \frac{26062}{111545} a^{4} + \frac{7737}{15935} a^{3} - \frac{841}{31870} a^{2} - \frac{290378}{557725} a - \frac{56988}{79675} \) (order $4$)
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:  \( 29499.8421143 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_5$ (as 12T74):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 7 conjugacy class representatives for $S_5$
Character table for $S_5$

Intermediate fields

\(\Q(\sqrt{-1}) \), 6.0.64000000.1

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

Sibling fields

Degree 5 sibling: data not computed
Degree 6 sibling: data not computed
Degree 10 siblings: data not computed
Degree 15 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.4.9.2$x^{4} - 2 x^{2} + 2$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.4.9.2$x^{4} - 2 x^{2} + 2$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.5.6.1$x^{5} + 10 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
5.5.6.1$x^{5} + 10 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$