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Properties

Label 13.13.3628385545...7424.2
Degree $13$
Signature $[13, 0]$
Discriminant $2^{36}\cdot 3^{16}\cdot 13^{6}\cdot 71^{4}$
Root discriminant $319.59$
Ramified primes $2, 3, 13, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $\PSL(3,3)$ (as 13T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256489920, 34006548, -257555808, -96770628, 27216624, 13437870, -1021200, -723404, 14832, 18444, -64, -222, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 222*x^11 - 64*x^10 + 18444*x^9 + 14832*x^8 - 723404*x^7 - 1021200*x^6 + 13437870*x^5 + 27216624*x^4 - 96770628*x^3 - 257555808*x^2 + 34006548*x + 256489920)
 
gp: K = bnfinit(x^13 - 222*x^11 - 64*x^10 + 18444*x^9 + 14832*x^8 - 723404*x^7 - 1021200*x^6 + 13437870*x^5 + 27216624*x^4 - 96770628*x^3 - 257555808*x^2 + 34006548*x + 256489920, 1)
 

Normalized defining polynomial

\( x^{13} - 222 x^{11} - 64 x^{10} + 18444 x^{9} + 14832 x^{8} - 723404 x^{7} - 1021200 x^{6} + 13437870 x^{5} + 27216624 x^{4} - 96770628 x^{3} - 257555808 x^{2} + 34006548 x + 256489920 \)

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

Invariants

Degree:  $13$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[13, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(362838554526023011241407675367424=2^{36}\cdot 3^{16}\cdot 13^{6}\cdot 71^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $319.59$
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:  $2, 3, 13, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6}$, $\frac{1}{6} a^{9} - \frac{1}{3} a^{6}$, $\frac{1}{18} a^{10} - \frac{1}{9} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{54} a^{11} + \frac{1}{54} a^{10} - \frac{1}{18} a^{9} - \frac{1}{27} a^{8} + \frac{2}{27} a^{7} - \frac{1}{3} a^{6} + \frac{1}{9} a^{5} - \frac{1}{3} a^{3} + \frac{2}{9} a^{2} - \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{867294893420992318975665118314084} a^{12} - \frac{2573665132922248773406859342777}{433647446710496159487832559157042} a^{11} + \frac{2316605094173640022738588692617}{433647446710496159487832559157042} a^{10} - \frac{7238522563530625266435025808624}{216823723355248079743916279578521} a^{9} + \frac{9742447069377023793609784220129}{216823723355248079743916279578521} a^{8} - \frac{14826611454586998733035218391350}{216823723355248079743916279578521} a^{7} - \frac{33522631018396226203376281624177}{72274574451749359914638759859507} a^{6} - \frac{2929795202375110659741469230997}{10324939207392765702091251408501} a^{5} - \frac{12654268957130522702157546838291}{48183049634499573276425839906338} a^{4} - \frac{18044323115413740675241044390476}{72274574451749359914638759859507} a^{3} + \frac{10843342680829441447147928507891}{72274574451749359914638759859507} a^{2} + \frac{26228690212332090607111275493342}{72274574451749359914638759859507} a - \frac{9752570052966867667552193091556}{24091524817249786638212919953169}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $12$
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:  \( 95129277252200 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\PSL(3,3)$ (as 13T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5616
The 12 conjugacy class representatives for $\PSL(3,3)$
Character table for $\PSL(3,3)$

Intermediate fields

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

Sibling fields

Degree 26 siblings: data not computed
Degree 39 siblings: data not computed
Arithmetically equvalently sibling: 13.13.362838554526023011241407675367424.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.13.0.1}{13} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{5}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.4.9.1$x^{4} + 6 x^{2} + 2$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.8.27.4$x^{8} + 4 x^{6} + 14 x^{4} + 16 x^{3} + 18$$8$$1$$27$$QD_{16}$$[2, 3, 7/2, 9/2]$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.3.2$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.6.9.15$x^{6} + 6 x^{4} + 6 x^{3} + 12$$6$$1$$9$$S_3\times C_3$$[3/2, 2]_{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
71.8.4.2$x^{8} - 357911 x^{2} + 279528491$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$