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Properties

Label 13.3.156408138976000.1
Degree $13$
Signature $[3, 5]$
Discriminant $-\,2^{8}\cdot 5^{3}\cdot 137\cdot 599\cdot 59561$
Root discriminant $12.36$
Ramified primes $2, 5, 137, 599, 59561$
Class number $1$
Class group Trivial
Galois group $S_{13}$ (as 13T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -12, 24, -36, 41, -35, 27, -20, 7, 4, -4, 3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 3*x^12 + 3*x^11 - 4*x^10 + 4*x^9 + 7*x^8 - 20*x^7 + 27*x^6 - 35*x^5 + 41*x^4 - 36*x^3 + 24*x^2 - 12*x + 4)
 
gp: K = bnfinit(x^13 - 3*x^12 + 3*x^11 - 4*x^10 + 4*x^9 + 7*x^8 - 20*x^7 + 27*x^6 - 35*x^5 + 41*x^4 - 36*x^3 + 24*x^2 - 12*x + 4, 1)
 

Normalized defining polynomial

\( x^{13} - 3 x^{12} + 3 x^{11} - 4 x^{10} + 4 x^{9} + 7 x^{8} - 20 x^{7} + 27 x^{6} - 35 x^{5} + 41 x^{4} - 36 x^{3} + 24 x^{2} - 12 x + 4 \)

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

Invariants

Degree:  $13$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-156408138976000=-\,2^{8}\cdot 5^{3}\cdot 137\cdot 599\cdot 59561\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.36$
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:  $2, 5, 137, 599, 59561$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{40} a^{11} - \frac{7}{40} a^{10} + \frac{17}{40} a^{9} - \frac{7}{20} a^{8} - \frac{9}{20} a^{7} - \frac{1}{8} a^{6} + \frac{3}{10} a^{5} + \frac{9}{40} a^{4} + \frac{1}{40} a^{3} - \frac{9}{40} a^{2} - \frac{7}{20} a + \frac{3}{20}$, $\frac{1}{1400} a^{12} - \frac{1}{280} a^{11} + \frac{363}{1400} a^{10} - \frac{19}{70} a^{9} - \frac{143}{700} a^{8} - \frac{121}{1400} a^{7} - \frac{239}{700} a^{6} + \frac{633}{1400} a^{5} + \frac{99}{1400} a^{4} + \frac{193}{1400} a^{3} - \frac{9}{175} a^{2} - \frac{13}{100} a - \frac{87}{350}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 127.407170199 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_{13}$ (as 13T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 6227020800
The 101 conjugacy class representatives for $S_{13}$ are not computed
Character table for $S_{13}$ is not computed

Intermediate fields

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

Sibling fields

Degree 26 sibling: 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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.11.0.1}{11} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.13.0.1}{13} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }$

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$$[2, 3]$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
$5$5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.9.0.1$x^{9} + x^{2} - 2 x + 2$$1$$9$$0$$C_9$$[\ ]^{9}$
$137$137.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
137.2.1.2$x^{2} + 411$$2$$1$$1$$C_2$$[\ ]_{2}$
137.9.0.1$x^{9} - 3 x + 23$$1$$9$$0$$C_9$$[\ ]^{9}$
599Data not computed
59561Data not computed