Properties

Label 7.3.14462809.1
Degree $7$
Signature $[3, 2]$
Discriminant $14462809$
Root discriminant $10.54$
Ramified prime $3803$
Class number $1$
Class group trivial
Galois group $\GL(3,2)$ (as 7T5)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 3*x^6 + 3*x^5 + 3*x^4 - 13*x^3 + 17*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^7 - 3*x^6 + 3*x^5 + 3*x^4 - 13*x^3 + 17*x^2 - 10*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 17, -13, 3, 3, -3, 1]);
 

\( x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 13 x^{3} + 17 x^{2} - 10 x + 1 \)

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

Invariants

Degree:  $7$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[3, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(14462809\)\(\medspace = 3803^{2}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $10.54$
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:  $3803$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$

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:  $4$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( a - 1 \),  \( \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + a^{3} - \frac{8}{5} a^{2} + \frac{6}{5} a + \frac{2}{5} \),  \( \frac{4}{5} a^{6} - \frac{9}{5} a^{5} + \frac{4}{5} a^{4} + 3 a^{3} - \frac{37}{5} a^{2} + \frac{34}{5} a - \frac{7}{5} \),  \( \frac{3}{5} a^{6} - \frac{3}{5} a^{5} - \frac{2}{5} a^{4} + 2 a^{3} - \frac{14}{5} a^{2} + \frac{3}{5} a + \frac{1}{5} \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 7.61729812148 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{3}\cdot(2\pi)^{2}\cdot 7.61729812148 \cdot 1}{2\sqrt{14462809}}\approx 0.316296477788$

Galois group

$\PSL(2,7)$ (as 7T5):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 168
The 6 conjugacy class representatives for $\GL(3,2)$
Character table for $\GL(3,2)$

Intermediate fields

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

Sibling fields

Degree 8 sibling: 8.0.209172844170481.1
Degree 14 siblings: Deg 14, Deg 14
Degree 21 sibling: Deg 21
Degree 24 sibling: Deg 24
Degree 28 sibling: Deg 28
Degree 42 siblings: Deg 42, Deg 42, Deg 42
Arithmetically equvalently sibling: 7.3.14462809.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.7.0.1}{7} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }$

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
3803Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
3.14462809.42t37.a.a$3$ $ 3803^{2}$ $x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 13 x^{3} + 17 x^{2} - 10 x + 1$ $\GL(3,2)$ (as 7T5) $0$ $-1$
3.14462809.42t37.a.b$3$ $ 3803^{2}$ $x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 13 x^{3} + 17 x^{2} - 10 x + 1$ $\GL(3,2)$ (as 7T5) $0$ $-1$
* 6.14462809.7t5.a.a$6$ $ 3803^{2}$ $x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 13 x^{3} + 17 x^{2} - 10 x + 1$ $\GL(3,2)$ (as 7T5) $1$ $2$
7.209...481.8t37.a.a$7$ $ 3803^{4}$ $x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 13 x^{3} + 17 x^{2} - 10 x + 1$ $\GL(3,2)$ (as 7T5) $1$ $-1$
8.209...481.21t14.a.a$8$ $ 3803^{4}$ $x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 13 x^{3} + 17 x^{2} - 10 x + 1$ $\GL(3,2)$ (as 7T5) $1$ $0$

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 *.