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Properties

Label 12.12.3932621360...1649.1
Degree $12$
Signature $[12, 0]$
Discriminant $13^{6}\cdot 59^{6}\cdot 3529^{6}$
Root discriminant $1645.22$
Ramified primes $13, 59, 3529$
Class number $20$ (GRH)
Class group $[2, 10]$ (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![2339838952, 3130627879, -1812709328, -456652720, 171815284, 15311536, -5362636, -200348, 72628, 1088, -443, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 443*x^10 + 1088*x^9 + 72628*x^8 - 200348*x^7 - 5362636*x^6 + 15311536*x^5 + 171815284*x^4 - 456652720*x^3 - 1812709328*x^2 + 3130627879*x + 2339838952)
 
gp: K = bnfinit(x^12 - 2*x^11 - 443*x^10 + 1088*x^9 + 72628*x^8 - 200348*x^7 - 5362636*x^6 + 15311536*x^5 + 171815284*x^4 - 456652720*x^3 - 1812709328*x^2 + 3130627879*x + 2339838952, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} - 443 x^{10} + 1088 x^{9} + 72628 x^{8} - 200348 x^{7} - 5362636 x^{6} + 15311536 x^{5} + 171815284 x^{4} - 456652720 x^{3} - 1812709328 x^{2} + 3130627879 x + 2339838952 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(393262136091127324453858365845510721649=13^{6}\cdot 59^{6}\cdot 3529^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1645.22$
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:  $13, 59, 3529$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{9} a^{3} + \frac{2}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{7} - \frac{1}{9} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{9} a^{8} + \frac{1}{9} a^{4} - \frac{2}{9}$, $\frac{1}{27} a^{9} + \frac{1}{27} a^{6} + \frac{1}{9} a^{4} + \frac{4}{27} a^{3} + \frac{1}{9} a^{2} + \frac{1}{9} a + \frac{1}{27}$, $\frac{1}{459} a^{10} + \frac{7}{459} a^{9} + \frac{4}{153} a^{8} + \frac{13}{459} a^{7} - \frac{20}{459} a^{6} - \frac{23}{459} a^{4} + \frac{7}{459} a^{3} - \frac{41}{153} a^{2} - \frac{41}{459} a + \frac{112}{459}$, $\frac{1}{18291344888712401844920699719617} a^{11} - \frac{2282329686074863369957084514}{18291344888712401844920699719617} a^{10} + \frac{332734726303209133519292014277}{18291344888712401844920699719617} a^{9} + \frac{826966286052671546296868664007}{18291344888712401844920699719617} a^{8} - \frac{569391948328815269421213009122}{18291344888712401844920699719617} a^{7} - \frac{209888159997943057869864018373}{18291344888712401844920699719617} a^{6} - \frac{758091938252790032739866462054}{18291344888712401844920699719617} a^{5} + \frac{168168983688908044378177568813}{1075961464041905990877688218801} a^{4} + \frac{1511031019557239705702937388220}{18291344888712401844920699719617} a^{3} + \frac{7123268274460200924773412702247}{18291344888712401844920699719617} a^{2} + \frac{4599797282548386391604222995324}{18291344888712401844920699719617} a + \frac{4501143700024539528051459184280}{18291344888712401844920699719617}$

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

Class group and class number

$C_{2}\times C_{10}$, 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:  $11$
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:  \( 506135353650000 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/5.11.0.1}{11} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.11.0.1}{11} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ R

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
$13$13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$59$59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3529Data not computed