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Properties

Label 12.0.76931173939...2209.1
Degree $12$
Signature $[0, 6]$
Discriminant $19^{6}\cdot 23^{6}\cdot 101^{10}$
Root discriminant $978.38$
Ramified primes $19, 23, 101$
Class number $80$ (GRH)
Class group $[2, 2, 20]$ (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![1084940875, -327865899, -81144849, 34602694, -799775, -1005130, 198008, -10924, -3778, 648, -15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 15*x^10 + 648*x^9 - 3778*x^8 - 10924*x^7 + 198008*x^6 - 1005130*x^5 - 799775*x^4 + 34602694*x^3 - 81144849*x^2 - 327865899*x + 1084940875)
 
gp: K = bnfinit(x^12 - 2*x^11 - 15*x^10 + 648*x^9 - 3778*x^8 - 10924*x^7 + 198008*x^6 - 1005130*x^5 - 799775*x^4 + 34602694*x^3 - 81144849*x^2 - 327865899*x + 1084940875, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} - 15 x^{10} + 648 x^{9} - 3778 x^{8} - 10924 x^{7} + 198008 x^{6} - 1005130 x^{5} - 799775 x^{4} + 34602694 x^{3} - 81144849 x^{2} - 327865899 x + 1084940875 \)

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:  \(769311739390426167457490080964882209=19^{6}\cdot 23^{6}\cdot 101^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $978.38$
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:  $19, 23, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{303} a^{6} - \frac{1}{303} a^{5} - \frac{8}{303} a^{4} + \frac{13}{303} a^{3} - \frac{30}{101} a^{2} + \frac{2}{101} a - \frac{118}{303}$, $\frac{1}{303} a^{7} - \frac{3}{101} a^{5} + \frac{5}{303} a^{4} - \frac{77}{303} a^{3} - \frac{28}{101} a^{2} - \frac{112}{303} a - \frac{118}{303}$, $\frac{1}{303} a^{8} - \frac{4}{303} a^{5} - \frac{149}{303} a^{4} + \frac{11}{101} a^{3} - \frac{13}{303} a^{2} - \frac{64}{303} a + \frac{50}{101}$, $\frac{1}{303} a^{9} + \frac{50}{101} a^{5} + \frac{1}{303} a^{4} + \frac{13}{101} a^{3} - \frac{121}{303} a^{2} - \frac{43}{101} a + \frac{134}{303}$, $\frac{1}{5151} a^{10} + \frac{8}{5151} a^{9} + \frac{2}{1717} a^{8} - \frac{2}{5151} a^{7} - \frac{4}{5151} a^{6} + \frac{115}{303} a^{5} - \frac{77}{1717} a^{4} - \frac{853}{5151} a^{3} - \frac{1388}{5151} a^{2} - \frac{467}{5151} a - \frac{2345}{5151}$, $\frac{1}{1367520393947239466391} a^{11} - \frac{16567508836257810}{455840131315746488797} a^{10} - \frac{37065964808441050}{26814125371514499341} a^{9} - \frac{104418377957124055}{455840131315746488797} a^{8} + \frac{230638450767058442}{1367520393947239466391} a^{7} - \frac{128444271133026179}{1367520393947239466391} a^{6} - \frac{170664779064848028506}{455840131315746488797} a^{5} - \frac{203325469808501623582}{455840131315746488797} a^{4} - \frac{637513580033938311931}{1367520393947239466391} a^{3} + \frac{186888321835260605300}{1367520393947239466391} a^{2} - \frac{54945147143693820908}{1367520393947239466391} a - \frac{397565692003048886471}{1367520393947239466391}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{20}$, which has order $80$ (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:  \( 357848365121 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ R R ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.11.0.1}{11} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$19$19.6.3.1$x^{6} - 38 x^{4} + 361 x^{2} - 109744$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.1$x^{6} - 38 x^{4} + 361 x^{2} - 109744$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$23$23.6.3.1$x^{6} - 46 x^{4} + 529 x^{2} - 194672$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.1$x^{6} - 46 x^{4} + 529 x^{2} - 194672$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$101$101.12.10.1$x^{12} - 1010 x^{6} + 7436529$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$