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Properties

Label 12.0.44968021684...7809.1
Degree $12$
Signature $[0, 6]$
Discriminant $5963263^{6}$
Root discriminant $2441.98$
Ramified prime $5963263$
Class number $20$ (GRH)
Class group $[20]$ (GRH)
Galois group $\PSL(2,11)$ (as 12T179)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![208327555, 299173773, 142645595, 39039918, 11374585, 1611514, 538148, 19996, 14438, -72, 193, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 193*x^10 - 72*x^9 + 14438*x^8 + 19996*x^7 + 538148*x^6 + 1611514*x^5 + 11374585*x^4 + 39039918*x^3 + 142645595*x^2 + 299173773*x + 208327555)
 
gp: K = bnfinit(x^12 - 2*x^11 + 193*x^10 - 72*x^9 + 14438*x^8 + 19996*x^7 + 538148*x^6 + 1611514*x^5 + 11374585*x^4 + 39039918*x^3 + 142645595*x^2 + 299173773*x + 208327555, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 193 x^{10} - 72 x^{9} + 14438 x^{8} + 19996 x^{7} + 538148 x^{6} + 1611514 x^{5} + 11374585 x^{4} + 39039918 x^{3} + 142645595 x^{2} + 299173773 x + 208327555 \)

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:  \(44968021684696030092752233382638572407809=5963263^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2441.98$
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:  $5963263$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{248702034649072267504660169577} a^{11} - \frac{27444677175861876267671590016}{248702034649072267504660169577} a^{10} - \frac{9671365120726058994851422564}{82900678216357422501553389859} a^{9} - \frac{21994440061506111560049911726}{248702034649072267504660169577} a^{8} - \frac{3056236685270706155420381286}{82900678216357422501553389859} a^{7} - \frac{1396246569909980414650763440}{248702034649072267504660169577} a^{6} + \frac{72284093237008459576173118354}{248702034649072267504660169577} a^{5} + \frac{42022078394290786909084081592}{248702034649072267504660169577} a^{4} + \frac{29925784257227644639049103937}{248702034649072267504660169577} a^{3} + \frac{95663234695389040545328347689}{248702034649072267504660169577} a^{2} + \frac{81411821141317252219619611127}{248702034649072267504660169577} a - \frac{23243466690092953785603848206}{248702034649072267504660169577}$

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

Class group and class number

$C_{20}$, which has order $20$ (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:  \( -1 \) (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:  \( 319144961169000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\PSL(2,11)$ (as 12T179):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 660
The 8 conjugacy class representatives for $\PSL(2,11)$
Character table for $\PSL(2,11)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 11 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 ${\href{/LocalNumberField/2.11.0.1}{11} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.11.0.1}{11} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5963263Data not computed