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Properties

Label 13.13.1162336843...2853.1
Degree $13$
Signature $[13, 0]$
Discriminant $13^{21}\cdot 19^{6}$
Root discriminant $245.27$
Ramified primes $13, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{13}:C_4$ (as 13T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![170991, 223587, -373113, -406770, 213772, 234988, -25922, -41431, 1339, 2951, -26, -91, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 91*x^11 - 26*x^10 + 2951*x^9 + 1339*x^8 - 41431*x^7 - 25922*x^6 + 234988*x^5 + 213772*x^4 - 406770*x^3 - 373113*x^2 + 223587*x + 170991)
 
gp: K = bnfinit(x^13 - 91*x^11 - 26*x^10 + 2951*x^9 + 1339*x^8 - 41431*x^7 - 25922*x^6 + 234988*x^5 + 213772*x^4 - 406770*x^3 - 373113*x^2 + 223587*x + 170991, 1)
 

Normalized defining polynomial

\( x^{13} - 91 x^{11} - 26 x^{10} + 2951 x^{9} + 1339 x^{8} - 41431 x^{7} - 25922 x^{6} + 234988 x^{5} + 213772 x^{4} - 406770 x^{3} - 373113 x^{2} + 223587 x + 170991 \)

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:  \(11623368434110587434575082172853=13^{21}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $245.27$
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:  $13, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{30} a^{9} + \frac{2}{15} a^{8} + \frac{1}{6} a^{7} - \frac{11}{30} a^{6} + \frac{4}{15} a^{5} + \frac{1}{3} a^{4} - \frac{1}{30} a^{3} + \frac{13}{30} a^{2} - \frac{1}{15} a + \frac{3}{10}$, $\frac{1}{30} a^{10} - \frac{1}{30} a^{8} - \frac{11}{30} a^{7} + \frac{1}{15} a^{6} - \frac{1}{15} a^{5} - \frac{1}{30} a^{4} + \frac{7}{30} a^{3} - \frac{7}{15} a^{2} + \frac{7}{30} a - \frac{1}{5}$, $\frac{1}{1530} a^{11} + \frac{4}{765} a^{9} + \frac{5}{306} a^{8} - \frac{133}{1530} a^{7} + \frac{499}{1530} a^{6} + \frac{431}{1530} a^{5} + \frac{427}{1530} a^{4} + \frac{217}{1530} a^{3} - \frac{13}{765} a^{2} - \frac{14}{255} a + \frac{9}{170}$, $\frac{1}{80933244956424090} a^{12} + \frac{6780400513957}{26977748318808030} a^{11} - \frac{139968025301179}{80933244956424090} a^{10} + \frac{27236638875587}{16186648991284818} a^{9} + \frac{4268572574649707}{80933244956424090} a^{8} + \frac{867192693874889}{4760779115083770} a^{7} + \frac{9486386166082297}{40466622478212045} a^{6} + \frac{8040150736366079}{40466622478212045} a^{5} - \frac{14827910485099139}{80933244956424090} a^{4} - \frac{2081578748917088}{8093324495642409} a^{3} + \frac{3605366453653294}{13488874159404015} a^{2} + \frac{1210123990451513}{4496291386468005} a + \frac{188776554642533}{2997527590978670}$

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:  \( 2019070466020 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{13}.C_2$ (as 13T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 52
The 7 conjugacy class representatives for $C_{13}:C_4$
Character table for $C_{13}:C_4$

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$13.13.21.2$x^{13} + 130 x^{9} + 13$$13$$1$$21$$C_{13}:C_4$$[7/4]_{4}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$