↑

Properties

Label 12.0.45521378559...9024.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{24}\cdot 3^{10}\cdot 11^{16}$
Root discriminant $244.45$
Ramified primes $2, 3, 11$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $\PSL(2,11)$ (as 12T179)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25818824, -1735376, 5326024, -615472, 443652, -61248, 7920, -528, 66, 220, -22, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 22*x^10 + 220*x^9 + 66*x^8 - 528*x^7 + 7920*x^6 - 61248*x^5 + 443652*x^4 - 615472*x^3 + 5326024*x^2 - 1735376*x + 25818824)
 
gp: K = bnfinit(x^12 - 4*x^11 - 22*x^10 + 220*x^9 + 66*x^8 - 528*x^7 + 7920*x^6 - 61248*x^5 + 443652*x^4 - 615472*x^3 + 5326024*x^2 - 1735376*x + 25818824, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} - 22 x^{10} + 220 x^{9} + 66 x^{8} - 528 x^{7} + 7920 x^{6} - 61248 x^{5} + 443652 x^{4} - 615472 x^{3} + 5326024 x^{2} - 1735376 x + 25818824 \)

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:  \(45521378559315317157500289024=2^{24}\cdot 3^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $244.45$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{48} a^{4} + \frac{1}{12} a^{3} - \frac{3}{8} a^{2} + \frac{5}{12} a + \frac{1}{24}$, $\frac{1}{144} a^{5} + \frac{7}{72} a^{3} + \frac{11}{36} a^{2} + \frac{1}{8} a + \frac{5}{18}$, $\frac{1}{144} a^{6} - \frac{1}{144} a^{4} - \frac{1}{9} a^{3} + \frac{7}{36} a - \frac{5}{24}$, $\frac{1}{432} a^{7} + \frac{1}{432} a^{6} + \frac{1}{108} a^{4} - \frac{31}{216} a^{3} - \frac{1}{24} a^{2} + \frac{1}{108} a - \frac{23}{108}$, $\frac{1}{13824} a^{8} + \frac{1}{1728} a^{7} - \frac{5}{3456} a^{6} + \frac{5}{1728} a^{5} + \frac{7}{1728} a^{4} + \frac{59}{432} a^{3} + \frac{653}{1728} a^{2} - \frac{85}{864} a - \frac{1655}{3456}$, $\frac{1}{41472} a^{9} - \frac{1}{41472} a^{8} - \frac{7}{10368} a^{7} - \frac{1}{10368} a^{6} - \frac{1}{2592} a^{5} + \frac{25}{5184} a^{4} - \frac{599}{5184} a^{3} - \frac{1871}{5184} a^{2} - \frac{3283}{10368} a - \frac{4289}{10368}$, $\frac{1}{41472} a^{10} + \frac{1}{41472} a^{8} + \frac{1}{2592} a^{7} + \frac{13}{10368} a^{6} - \frac{7}{5184} a^{5} - \frac{7}{1296} a^{4} + \frac{277}{2592} a^{3} + \frac{4651}{10368} a^{2} - \frac{7}{288} a - \frac{2579}{10368}$, $\frac{1}{270107136} a^{11} + \frac{5}{1875744} a^{10} + \frac{109}{135053568} a^{9} + \frac{917}{30011904} a^{8} + \frac{6359}{11254464} a^{7} - \frac{34159}{22508928} a^{6} + \frac{148}{175851} a^{5} + \frac{101195}{11254464} a^{4} - \frac{547499}{7502976} a^{3} - \frac{14552687}{33763392} a^{2} + \frac{4672415}{11254464} a + \frac{8756845}{67526784}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 11589679277.7 \) (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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.11.0.1}{11} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.11.10.1$x^{11} - 3$$11$$1$$10$$C_{11}:C_5$$[\ ]_{11}^{5}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.11.16.5$x^{11} + 88 x^{6} + 11$$11$$1$$16$$C_{11}:C_5$$[8/5]_{5}$