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Properties

Label 10.10.6043755034...7881.1
Degree $10$
Signature $[10, 0]$
Discriminant $7^{6}\cdot 283^{6}$
Root discriminant $95.09$
Ramified primes $7, 283$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $\PSL(2,9)$ (as 10T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-115, -1243, 568, 3042, -1392, -806, 375, 71, -34, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 - 34*x^8 + 71*x^7 + 375*x^6 - 806*x^5 - 1392*x^4 + 3042*x^3 + 568*x^2 - 1243*x - 115)
 
gp: K = bnfinit(x^10 - 2*x^9 - 34*x^8 + 71*x^7 + 375*x^6 - 806*x^5 - 1392*x^4 + 3042*x^3 + 568*x^2 - 1243*x - 115, 1)
 

Normalized defining polynomial

\( x^{10} - 2 x^{9} - 34 x^{8} + 71 x^{7} + 375 x^{6} - 806 x^{5} - 1392 x^{4} + 3042 x^{3} + 568 x^{2} - 1243 x - 115 \)

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

Invariants

Degree:  $10$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(60437550349593857881=7^{6}\cdot 283^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.09$
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:  $7, 283$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{212681453} a^{9} + \frac{102869251}{212681453} a^{8} - \frac{87137587}{212681453} a^{7} - \frac{49410846}{212681453} a^{6} - \frac{68356}{212681453} a^{5} - \frac{56459788}{212681453} a^{4} - \frac{61970454}{212681453} a^{3} + \frac{10367340}{212681453} a^{2} + \frac{9401738}{212681453} a - \frac{1226259}{212681453}$

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:  $9$
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:  \( 17556944.4362 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_6$ (as 10T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 360
The 7 conjugacy class representatives for $\PSL(2,9)$
Character table for $\PSL(2,9)$

Intermediate fields

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

Sibling fields

Degree 6 siblings: 6.6.15400609258321.1, 6.6.192293689.1
Degree 15 siblings: 15.15.11621759510846642583679213009.1, Deg 15
Degree 20 sibling: Deg 20
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 sibling: data not computed
Degree 45 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.4.3.2$x^{4} - 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
283Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
5.7e4_283e2.6t15.2c1$5$ $ 7^{4} \cdot 283^{2}$ $x^{10} - 2 x^{9} - 34 x^{8} + 71 x^{7} + 375 x^{6} - 806 x^{5} - 1392 x^{4} + 3042 x^{3} + 568 x^{2} - 1243 x - 115$ $\PSL(2,9)$ (as 10T26) $1$ $5$
5.7e4_283e4.6t15.2c1$5$ $ 7^{4} \cdot 283^{4}$ $x^{10} - 2 x^{9} - 34 x^{8} + 71 x^{7} + 375 x^{6} - 806 x^{5} - 1392 x^{4} + 3042 x^{3} + 568 x^{2} - 1243 x - 115$ $\PSL(2,9)$ (as 10T26) $1$ $5$
8.7e6_283e6.36t555.2c1$8$ $ 7^{6} \cdot 283^{6}$ $x^{10} - 2 x^{9} - 34 x^{8} + 71 x^{7} + 375 x^{6} - 806 x^{5} - 1392 x^{4} + 3042 x^{3} + 568 x^{2} - 1243 x - 115$ $\PSL(2,9)$ (as 10T26) $1$ $8$
8.7e6_283e6.36t555.2c2$8$ $ 7^{6} \cdot 283^{6}$ $x^{10} - 2 x^{9} - 34 x^{8} + 71 x^{7} + 375 x^{6} - 806 x^{5} - 1392 x^{4} + 3042 x^{3} + 568 x^{2} - 1243 x - 115$ $\PSL(2,9)$ (as 10T26) $1$ $8$
* 9.7e6_283e6.10t26.2c1$9$ $ 7^{6} \cdot 283^{6}$ $x^{10} - 2 x^{9} - 34 x^{8} + 71 x^{7} + 375 x^{6} - 806 x^{5} - 1392 x^{4} + 3042 x^{3} + 568 x^{2} - 1243 x - 115$ $\PSL(2,9)$ (as 10T26) $1$ $9$
10.7e8_283e6.30t88.2c1$10$ $ 7^{8} \cdot 283^{6}$ $x^{10} - 2 x^{9} - 34 x^{8} + 71 x^{7} + 375 x^{6} - 806 x^{5} - 1392 x^{4} + 3042 x^{3} + 568 x^{2} - 1243 x - 115$ $\PSL(2,9)$ (as 10T26) $1$ $10$

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 *.