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Properties

Label 13.13.3628385545...7424.1
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![-132080544, -240370524, -108639936, 30740796, 30420288, 2053638, -2357568, -390420, 63024, 15084, -528, -214, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 214*x^11 - 528*x^10 + 15084*x^9 + 63024*x^8 - 390420*x^7 - 2357568*x^6 + 2053638*x^5 + 30420288*x^4 + 30740796*x^3 - 108639936*x^2 - 240370524*x - 132080544)
 
gp: K = bnfinit(x^13 - 214*x^11 - 528*x^10 + 15084*x^9 + 63024*x^8 - 390420*x^7 - 2357568*x^6 + 2053638*x^5 + 30420288*x^4 + 30740796*x^3 - 108639936*x^2 - 240370524*x - 132080544, 1)
 

Normalized defining polynomial

\( x^{13} - 214 x^{11} - 528 x^{10} + 15084 x^{9} + 63024 x^{8} - 390420 x^{7} - 2357568 x^{6} + 2053638 x^{5} + 30420288 x^{4} + 30740796 x^{3} - 108639936 x^{2} - 240370524 x - 132080544 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5}$, $\frac{1}{6} a^{8} + \frac{1}{3} a^{6}$, $\frac{1}{6} a^{9} + \frac{1}{3} a^{5}$, $\frac{1}{6} a^{10} + \frac{1}{3} a^{6}$, $\frac{1}{18} a^{11} - \frac{1}{18} a^{9} - \frac{1}{3} a^{5}$, $\frac{1}{142345197933439670031328369951860} a^{12} + \frac{479873522506100277450120696989}{23724199655573278338554728325310} a^{11} - \frac{2190603316872683181894658187737}{35586299483359917507832092487965} a^{10} + \frac{52357581949050168476539288633}{790806655185775944618490944177} a^{9} - \frac{226304979526968055102818606206}{3954033275928879723092454720885} a^{8} - \frac{315233711945293812675091358771}{2372419965557327833855472832531} a^{7} - \frac{10380977817418439985117938062}{263602218395258648206163648059} a^{6} - \frac{522306352393938022067135684473}{3954033275928879723092454720885} a^{5} + \frac{213944768844053931367512317977}{7908066551857759446184909441770} a^{4} + \frac{1092073683964977805362526001407}{3954033275928879723092454720885} a^{3} + \frac{314785579750768888199998214198}{1318011091976293241030818240295} a^{2} - \frac{390690417404376266892120912698}{790806655185775944618490944177} a - \frac{621637902947931683216395902868}{1318011091976293241030818240295}$

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: 13.13.362838554526023011241407675367424.2

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$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.9.15.27$x^{9} + 6 x^{8} + 6 x^{7} + 3 x^{3} + 3$$9$$1$$15$$S_3\times C_3$$[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}$