Properties

Label 13.1.489...249.1
Degree $13$
Signature $[1, 6]$
Discriminant $4.892\times 10^{20}$
Root discriminant $39.04$
Ramified primes $7, 401$
Class number $1$
Class group trivial
Galois group $D_{13}$ (as 13T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 3*x^12 + 8*x^11 - 23*x^10 - 10*x^9 + 14*x^8 + 98*x^7 + 269*x^6 + 309*x^5 + 121*x^4 + 175*x^3 - 28*x^2 + 245*x + 49)
 
gp: K = bnfinit(x^13 - 3*x^12 + 8*x^11 - 23*x^10 - 10*x^9 + 14*x^8 + 98*x^7 + 269*x^6 + 309*x^5 + 121*x^4 + 175*x^3 - 28*x^2 + 245*x + 49, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, 245, -28, 175, 121, 309, 269, 98, 14, -10, -23, 8, -3, 1]);
 

\(x^{13} - 3 x^{12} + 8 x^{11} - 23 x^{10} - 10 x^{9} + 14 x^{8} + 98 x^{7} + 269 x^{6} + 309 x^{5} + 121 x^{4} + 175 x^{3} - 28 x^{2} + 245 x + 49\)  Toggle raw display

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

Invariants

Degree:  $13$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(489163986649360075249\)\(\medspace = 7^{6}\cdot 401^{6}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $39.04$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $7, 401$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{8} - \frac{2}{7} a^{7} + \frac{2}{7} a^{6} + \frac{3}{7} a^{5} - \frac{3}{7} a^{4} - \frac{2}{7} a^{3} + \frac{2}{7} a^{2}$, $\frac{1}{7} a^{10} - \frac{3}{7} a^{8} - \frac{2}{7} a^{6} + \frac{2}{7} a^{4} + \frac{2}{7} a^{2}$, $\frac{1}{427} a^{11} - \frac{29}{427} a^{10} - \frac{16}{427} a^{9} - \frac{103}{427} a^{8} + \frac{171}{427} a^{7} + \frac{123}{427} a^{6} + \frac{75}{427} a^{5} - \frac{19}{427} a^{4} + \frac{14}{61} a^{3} - \frac{4}{61} a^{2} - \frac{27}{61} a + \frac{28}{61}$, $\frac{1}{675202881395} a^{12} - \frac{11723827}{11068899695} a^{11} - \frac{5377361124}{675202881395} a^{10} + \frac{6819782229}{96457554485} a^{9} - \frac{142788197757}{675202881395} a^{8} + \frac{21593619867}{675202881395} a^{7} + \frac{7545472259}{135040576279} a^{6} + \frac{56710326179}{675202881395} a^{5} - \frac{6154245252}{675202881395} a^{4} + \frac{182041008699}{675202881395} a^{3} + \frac{310567263429}{675202881395} a^{2} + \frac{25170688678}{96457554485} a + \frac{9180320618}{96457554485}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $6$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 297132.5632297882 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{6}\cdot 297132.5632297882 \cdot 1}{2\sqrt{489163986649360075249}}\approx 0.826612982119755$

Galois group

$D_{13}$ (as 13T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 26
The 8 conjugacy class representatives for $D_{13}$
Character table for $D_{13}$

Intermediate fields

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

Sibling fields

Galois closure: Deg 26

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.13.0.1}{13} }$ ${\href{/padicField/3.13.0.1}{13} }$ ${\href{/padicField/5.2.0.1}{2} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }$ R ${\href{/padicField/11.13.0.1}{13} }$ ${\href{/padicField/13.13.0.1}{13} }$ ${\href{/padicField/17.13.0.1}{13} }$ ${\href{/padicField/19.13.0.1}{13} }$ ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.13.0.1}{13} }$ ${\href{/padicField/31.13.0.1}{13} }$ ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.13.0.1}{13} }$ ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.13.0.1}{13} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
401Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2807.2t1.a.a$1$ $ 7 \cdot 401 $ \(\Q(\sqrt{-2807}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.2807.13t2.a.f$2$ $ 7 \cdot 401 $ 13.1.489163986649360075249.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.2807.13t2.a.b$2$ $ 7 \cdot 401 $ 13.1.489163986649360075249.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.2807.13t2.a.e$2$ $ 7 \cdot 401 $ 13.1.489163986649360075249.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.2807.13t2.a.c$2$ $ 7 \cdot 401 $ 13.1.489163986649360075249.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.2807.13t2.a.a$2$ $ 7 \cdot 401 $ 13.1.489163986649360075249.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.2807.13t2.a.d$2$ $ 7 \cdot 401 $ 13.1.489163986649360075249.1 $D_{13}$ (as 13T2) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.