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Properties

Label 10.10.1542999470...9877.1
Degree $10$
Signature $[10, 0]$
Discriminant $3^{2}\cdot 19^{2}\cdot 37^{5}\cdot 2617^{2}$
Root discriminant $65.89$
Ramified primes $3, 19, 37, 2617$
Class number $2$
Class group $[2]$
Galois group $S_5\times C_2$ (as 10T22)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10519, -4289, 17924, 4380, -8188, -691, 1147, 46, -59, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 - 59*x^8 + 46*x^7 + 1147*x^6 - 691*x^5 - 8188*x^4 + 4380*x^3 + 17924*x^2 - 4289*x - 10519)
 
gp: K = bnfinit(x^10 - x^9 - 59*x^8 + 46*x^7 + 1147*x^6 - 691*x^5 - 8188*x^4 + 4380*x^3 + 17924*x^2 - 4289*x - 10519, 1)
 

Normalized defining polynomial

\( x^{10} - x^{9} - 59 x^{8} + 46 x^{7} + 1147 x^{6} - 691 x^{5} - 8188 x^{4} + 4380 x^{3} + 17924 x^{2} - 4289 x - 10519 \)

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:  \(1542999470252189877=3^{2}\cdot 19^{2}\cdot 37^{5}\cdot 2617^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.89$
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:  $3, 19, 37, 2617$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{551345444446493} a^{9} - \frac{108962574078232}{551345444446493} a^{8} + \frac{149697503215288}{551345444446493} a^{7} - \frac{237684886440730}{551345444446493} a^{6} - \frac{83109865705344}{551345444446493} a^{5} - \frac{83603233083532}{551345444446493} a^{4} + \frac{193133740687798}{551345444446493} a^{3} + \frac{104705135941568}{551345444446493} a^{2} - \frac{252767213942436}{551345444446493} a + \frac{71563228037303}{551345444446493}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 732740.204608 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_5$ (as 10T22):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 240
The 14 conjugacy class representatives for $S_5\times C_2$
Character table for $S_5\times C_2$

Intermediate fields

\(\Q(\sqrt{37}) \), 5.5.149169.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 sibling: data not computed
Degree 12 siblings: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 30 siblings: data not computed
Degree 40 siblings: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }$ R ${\href{/LocalNumberField/5.10.0.1}{10} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ R ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37Data not computed
2617Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.3_19_37_2617.2t1.1c1$1$ $ 3 \cdot 19 \cdot 37 \cdot 2617 $ $x^{2} - x - 1379813$ $C_2$ (as 2T1) $1$ $1$
1.3_19_2617.2t1.1c1$1$ $ 3 \cdot 19 \cdot 2617 $ $x^{2} - x - 37292$ $C_2$ (as 2T1) $1$ $1$
* 1.37.2t1.1c1$1$ $ 37 $ $x^{2} - x - 9$ $C_2$ (as 2T1) $1$ $1$
4.3e3_19e3_37e4_2617e3.10t22.2c1$4$ $ 3^{3} \cdot 19^{3} \cdot 37^{4} \cdot 2617^{3}$ $x^{10} - x^{9} - 59 x^{8} + 46 x^{7} + 1147 x^{6} - 691 x^{5} - 8188 x^{4} + 4380 x^{3} + 17924 x^{2} - 4289 x - 10519$ $S_5\times C_2$ (as 10T22) $1$ $4$
* 4.3_19_37e4_2617.10t22.2c1$4$ $ 3 \cdot 19 \cdot 37^{4} \cdot 2617 $ $x^{10} - x^{9} - 59 x^{8} + 46 x^{7} + 1147 x^{6} - 691 x^{5} - 8188 x^{4} + 4380 x^{3} + 17924 x^{2} - 4289 x - 10519$ $S_5\times C_2$ (as 10T22) $1$ $4$
* 4.3_19_2617.5t5.1c1$4$ $ 3 \cdot 19 \cdot 2617 $ $x^{5} - 6 x^{3} - 3 x^{2} + 4 x + 1$ $S_5$ (as 5T5) $1$ $4$
4.3e3_19e3_2617e3.10t12.1c1$4$ $ 3^{3} \cdot 19^{3} \cdot 2617^{3}$ $x^{5} - 6 x^{3} - 3 x^{2} + 4 x + 1$ $S_5$ (as 5T5) $1$ $4$
5.3e2_19e2_37e5_2617e2.12t123.2c1$5$ $ 3^{2} \cdot 19^{2} \cdot 37^{5} \cdot 2617^{2}$ $x^{10} - x^{9} - 59 x^{8} + 46 x^{7} + 1147 x^{6} - 691 x^{5} - 8188 x^{4} + 4380 x^{3} + 17924 x^{2} - 4289 x - 10519$ $S_5\times C_2$ (as 10T22) $1$ $5$
5.3e3_19e3_37e5_2617e3.12t123.2c1$5$ $ 3^{3} \cdot 19^{3} \cdot 37^{5} \cdot 2617^{3}$ $x^{10} - x^{9} - 59 x^{8} + 46 x^{7} + 1147 x^{6} - 691 x^{5} - 8188 x^{4} + 4380 x^{3} + 17924 x^{2} - 4289 x - 10519$ $S_5\times C_2$ (as 10T22) $1$ $5$
5.3e3_19e3_2617e3.6t14.1c1$5$ $ 3^{3} \cdot 19^{3} \cdot 2617^{3}$ $x^{5} - 6 x^{3} - 3 x^{2} + 4 x + 1$ $S_5$ (as 5T5) $1$ $5$
5.3e2_19e2_2617e2.10t13.1c1$5$ $ 3^{2} \cdot 19^{2} \cdot 2617^{2}$ $x^{5} - 6 x^{3} - 3 x^{2} + 4 x + 1$ $S_5$ (as 5T5) $1$ $5$
6.3e3_19e3_2617e3.20t35.1c1$6$ $ 3^{3} \cdot 19^{3} \cdot 2617^{3}$ $x^{5} - 6 x^{3} - 3 x^{2} + 4 x + 1$ $S_5$ (as 5T5) $1$ $6$
6.3e3_19e3_37e6_2617e3.20t65.2c1$6$ $ 3^{3} \cdot 19^{3} \cdot 37^{6} \cdot 2617^{3}$ $x^{10} - x^{9} - 59 x^{8} + 46 x^{7} + 1147 x^{6} - 691 x^{5} - 8188 x^{4} + 4380 x^{3} + 17924 x^{2} - 4289 x - 10519$ $S_5\times C_2$ (as 10T22) $1$ $6$

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 *.