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Properties

Label 12.0.98032111360...8896.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{16}\cdot 11^{18}$
Root discriminant $315.70$
Ramified primes $2, 3, 11$
Class number $12$ (GRH)
Class group $[12]$ (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![8504001, -12431826, 8409258, -3637986, 1140183, -293304, 67760, -12342, 2178, -264, 66, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 66*x^10 - 264*x^9 + 2178*x^8 - 12342*x^7 + 67760*x^6 - 293304*x^5 + 1140183*x^4 - 3637986*x^3 + 8409258*x^2 - 12431826*x + 8504001)
 
gp: K = bnfinit(x^12 + 66*x^10 - 264*x^9 + 2178*x^8 - 12342*x^7 + 67760*x^6 - 293304*x^5 + 1140183*x^4 - 3637986*x^3 + 8409258*x^2 - 12431826*x + 8504001, 1)
 

Normalized defining polynomial

\( x^{12} + 66 x^{10} - 264 x^{9} + 2178 x^{8} - 12342 x^{7} + 67760 x^{6} - 293304 x^{5} + 1140183 x^{4} - 3637986 x^{3} + 8409258 x^{2} - 12431826 x + 8504001 \)

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:  \(980321113608067580846177488896=2^{12}\cdot 3^{16}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $315.70$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2}$, $\frac{1}{12} a^{9} - \frac{1}{12} a^{7} - \frac{1}{12} a^{6} - \frac{1}{4} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{36} a^{10} - \frac{1}{36} a^{9} + \frac{1}{36} a^{8} + \frac{1}{18} a^{7} - \frac{2}{9} a^{6} - \frac{7}{36} a^{5} - \frac{1}{2} a^{4} - \frac{5}{12} a^{3} - \frac{1}{6} a^{2} + \frac{1}{4}$, $\frac{1}{6390012980039653969969140} a^{11} + \frac{1238112367049364534091}{580910270912695815451740} a^{10} + \frac{7246276403067943761541}{290455135456347907725870} a^{9} + \frac{5460348707196593793719}{290455135456347907725870} a^{8} - \frac{30234866623405919201959}{580910270912695815451740} a^{7} + \frac{39974348895321417086267}{290455135456347907725870} a^{6} + \frac{2643325435086408362621}{64545585656966201716860} a^{5} + \frac{6573213712283202645929}{38727351394179721030116} a^{4} + \frac{22408147217897206505549}{48409189242724651287645} a^{3} - \frac{2607255286323807157359}{21515195218988733905620} a^{2} + \frac{289021734304603658193}{1075759760949436695281} a - \frac{2597602313224026089323}{21515195218988733905620}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( 8088983337.07 \) (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 R R ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.11.0.1}{11} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$2$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.11.18.5$x^{11} + 110 x^{8} + 11$$11$$1$$18$$C_{11}:C_5$$[9/5]_{5}$