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Properties

Label 12.0.46506419261...3125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 47^{8}$
Root discriminant $43.55$
Ramified primes $5, 47$
Class number $16$ (GRH)
Class group $[2, 2, 2, 2]$ (GRH)
Galois group $C_3 : C_4$ (as 12T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2025, 18225, 67590, 63330, 57661, 19242, 5347, 635, 311, -25, 23, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 23*x^10 - 25*x^9 + 311*x^8 + 635*x^7 + 5347*x^6 + 19242*x^5 + 57661*x^4 + 63330*x^3 + 67590*x^2 + 18225*x + 2025)
 
gp: K = bnfinit(x^12 - 4*x^11 + 23*x^10 - 25*x^9 + 311*x^8 + 635*x^7 + 5347*x^6 + 19242*x^5 + 57661*x^4 + 63330*x^3 + 67590*x^2 + 18225*x + 2025, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 23 x^{10} - 25 x^{9} + 311 x^{8} + 635 x^{7} + 5347 x^{6} + 19242 x^{5} + 57661 x^{4} + 63330 x^{3} + 67590 x^{2} + 18225 x + 2025 \)

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:  \(46506419261251953125=5^{9}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.55$
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:  $5, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{90} a^{10} - \frac{2}{45} a^{9} - \frac{11}{45} a^{8} + \frac{2}{9} a^{7} - \frac{2}{45} a^{6} + \frac{1}{18} a^{5} - \frac{4}{45} a^{4} - \frac{1}{5} a^{3} + \frac{8}{45} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{7532581448463079593089070} a^{11} - \frac{26328500861294511427309}{7532581448463079593089070} a^{10} + \frac{271417949848669676736563}{7532581448463079593089070} a^{9} + \frac{102618853886899214180888}{753258144846307959308907} a^{8} - \frac{200644443360208122443569}{7532581448463079593089070} a^{7} + \frac{286118246978690679025793}{753258144846307959308907} a^{6} + \frac{153176855345809645945027}{7532581448463079593089070} a^{5} - \frac{141574044954279150982373}{1255430241410513265514845} a^{4} - \frac{115142457372311571913739}{7532581448463079593089070} a^{3} - \frac{79884697231764006020047}{502172096564205306205938} a^{2} - \frac{10425699337129946317037}{55796899618245034022882} a - \frac{9911218209040745736633}{27898449809122517011441}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (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:  \( -\frac{578325506706052573739}{3766290724231539796544535} a^{11} + \frac{2326910866951547315063}{3766290724231539796544535} a^{10} - \frac{2682470516650234348973}{753258144846307959308907} a^{9} + \frac{15122809888337690940851}{3766290724231539796544535} a^{8} - \frac{182383815238131816830734}{3766290724231539796544535} a^{7} - \frac{356160427525064702645008}{3766290724231539796544535} a^{6} - \frac{3116011599425457238144838}{3766290724231539796544535} a^{5} - \frac{1225008335807160056220331}{418476747136837755171615} a^{4} - \frac{6683747301160612132825717}{753258144846307959308907} a^{3} - \frac{12075113062086495501299461}{1255430241410513265514845} a^{2} - \frac{891792436704481755086518}{83695349427367551034323} a - \frac{52866745535577643542557}{27898449809122517011441} \) (order $10$)
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:  \( 22045.6143754 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:C_4$ (as 12T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12
The 6 conjugacy class representatives for $C_3 : C_4$
Character table for $C_3 : C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.11045.1 x3, \(\Q(\zeta_{5})\), 6.6.609960125.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$47$47.12.8.1$x^{12} - 141 x^{9} + 6627 x^{6} - 103823 x^{3} + 289457797239$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$