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Properties

Label 13.13.3087672181...5625.1
Degree $13$
Signature $[13, 0]$
Discriminant $3^{28}\cdot 5^{18}\cdot 29^{12}$
Root discriminant $2214.91$
Ramified primes $3, 5, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_{13}$ (as 13T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1362894158, 1350962689, 1061716068, -502327474, -126847180, 48766779, 5276958, -1919784, -94473, 35420, 737, -306, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 2*x^12 - 306*x^11 + 737*x^10 + 35420*x^9 - 94473*x^8 - 1919784*x^7 + 5276958*x^6 + 48766779*x^5 - 126847180*x^4 - 502327474*x^3 + 1061716068*x^2 + 1350962689*x - 1362894158)
 
gp: K = bnfinit(x^13 - 2*x^12 - 306*x^11 + 737*x^10 + 35420*x^9 - 94473*x^8 - 1919784*x^7 + 5276958*x^6 + 48766779*x^5 - 126847180*x^4 - 502327474*x^3 + 1061716068*x^2 + 1350962689*x - 1362894158, 1)
 

Normalized defining polynomial

\( x^{13} - 2 x^{12} - 306 x^{11} + 737 x^{10} + 35420 x^{9} - 94473 x^{8} - 1919784 x^{7} + 5276958 x^{6} + 48766779 x^{5} - 126847180 x^{4} - 502327474 x^{3} + 1061716068 x^{2} + 1350962689 x - 1362894158 \)

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:  \(30876721812776703244552125405887603759765625=3^{28}\cdot 5^{18}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2214.91$
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:  $3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{25} a^{4} - \frac{1}{25} a^{3} + \frac{12}{25} a^{2} + \frac{1}{25} a + \frac{12}{25}$, $\frac{1}{125} a^{5} + \frac{11}{125} a^{3} - \frac{62}{125} a^{2} - \frac{62}{125} a + \frac{12}{125}$, $\frac{1}{625} a^{6} + \frac{1}{625} a^{5} + \frac{11}{625} a^{4} - \frac{51}{625} a^{3} + \frac{1}{625} a^{2} - \frac{12}{25} a + \frac{262}{625}$, $\frac{1}{3125} a^{7} + \frac{2}{3125} a^{6} + \frac{12}{3125} a^{5} - \frac{8}{625} a^{4} - \frac{2}{125} a^{3} - \frac{924}{3125} a^{2} + \frac{587}{3125} a + \frac{1512}{3125}$, $\frac{1}{15625} a^{8} - \frac{2}{15625} a^{7} + \frac{4}{15625} a^{6} + \frac{37}{15625} a^{5} + \frac{22}{3125} a^{4} + \frac{651}{15625} a^{3} + \frac{5908}{15625} a^{2} - \frac{2336}{15625} a - \frac{7673}{15625}$, $\frac{1}{78125} a^{9} - \frac{1}{78125} a^{8} + \frac{2}{78125} a^{7} + \frac{41}{78125} a^{6} + \frac{147}{78125} a^{5} + \frac{761}{78125} a^{4} + \frac{6559}{78125} a^{3} + \frac{3572}{78125} a^{2} + \frac{5616}{78125} a + \frac{7952}{78125}$, $\frac{1}{1171875} a^{10} - \frac{1}{234375} a^{9} + \frac{2}{390625} a^{8} + \frac{11}{390625} a^{7} - \frac{214}{390625} a^{6} + \frac{891}{390625} a^{5} - \frac{224}{78125} a^{4} - \frac{11513}{390625} a^{3} + \frac{140651}{390625} a^{2} - \frac{333262}{1171875} a + \frac{388817}{1171875}$, $\frac{1}{5859375} a^{11} + \frac{1}{5859375} a^{10} - \frac{8}{1953125} a^{9} + \frac{23}{1953125} a^{8} - \frac{148}{1953125} a^{7} - \frac{393}{1953125} a^{6} + \frac{4226}{1953125} a^{5} - \frac{18233}{1953125} a^{4} + \frac{71573}{1953125} a^{3} - \frac{1317169}{5859375} a^{2} - \frac{322151}{1171875} a - \frac{3616}{1953125}$, $\frac{1}{146484375} a^{12} + \frac{4}{48828125} a^{11} - \frac{13}{146484375} a^{10} - \frac{38}{9765625} a^{9} + \frac{171}{9765625} a^{8} + \frac{5854}{48828125} a^{7} - \frac{15222}{48828125} a^{6} - \frac{41997}{48828125} a^{5} - \frac{96698}{9765625} a^{4} - \frac{1801352}{29296875} a^{3} - \frac{12064288}{48828125} a^{2} - \frac{60674528}{146484375} a + \frac{18595599}{48828125}$

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:  \( 4344433704670000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_{13}$ (as 13T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3113510400
The 55 conjugacy class representatives for $A_{13}$ are not computed
Character table for $A_{13}$ is not computed

Intermediate fields

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ R R ${\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.13.0.1}{13} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ R ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.13.0.1}{13} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.3.5.2$x^{3} + 21$$3$$1$$5$$S_3$$[5/2]_{2}$
3.9.23.38$x^{9} + 9 x^{8} + 6 x^{6} + 3$$9$$1$$23$$(C_3^2:C_3):C_2$$[2, 5/2, 17/6, 19/6]_{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.10.18.23$x^{10} - 20 x^{9} + 105$$10$$1$$18$$(C_5^2 : C_8):C_2$$[17/8, 17/8]_{8}^{2}$
$29$29.13.12.1$x^{13} - 29$$13$$1$$12$$C_{13}:C_3$$[\ ]_{13}^{3}$