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Properties

Label 10.2.20272865295...5625.1
Degree $10$
Signature $[2, 4]$
Discriminant $3^{12}\cdot 5^{18}$
Root discriminant $67.72$
Ramified primes $3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $M_{10}$ (as 10T31)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8472, 3345, 5625, -3780, 480, 39, -15, 30, 0, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 5*x^9 + 30*x^7 - 15*x^6 + 39*x^5 + 480*x^4 - 3780*x^3 + 5625*x^2 + 3345*x - 8472)
 
gp: K = bnfinit(x^10 - 5*x^9 + 30*x^7 - 15*x^6 + 39*x^5 + 480*x^4 - 3780*x^3 + 5625*x^2 + 3345*x - 8472, 1)
 

Normalized defining polynomial

\( x^{10} - 5 x^{9} + 30 x^{7} - 15 x^{6} + 39 x^{5} + 480 x^{4} - 3780 x^{3} + 5625 x^{2} + 3345 x - 8472 \)

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

Invariants

Degree:  $10$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2027286529541015625=3^{12}\cdot 5^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.72$
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:  $3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{20} a^{7} + \frac{1}{20} a^{6} - \frac{1}{20} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{20} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{40} a^{8} + \frac{3}{40} a^{6} + \frac{1}{40} a^{5} - \frac{1}{8} a^{4} - \frac{19}{40} a^{3} - \frac{1}{4} a^{2} + \frac{13}{40} a + \frac{2}{5}$, $\frac{1}{22264323400} a^{9} + \frac{85423237}{22264323400} a^{8} - \frac{477851391}{22264323400} a^{7} + \frac{641584507}{5566080850} a^{6} - \frac{636636513}{2783040425} a^{5} - \frac{970378961}{5566080850} a^{4} - \frac{41990493}{22264323400} a^{3} + \frac{3804990219}{22264323400} a^{2} - \frac{1655363957}{22264323400} a + \frac{869751917}{2783040425}$

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:  $5$
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:  \( 1792451.54077 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$M_{10}$ (as 10T31):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 720
The 8 conjugacy class representatives for $M_{10}$
Character table for $M_{10}$

Intermediate fields

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

Sibling fields

Degree 12 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.9.12.21$x^{9} + 3 x^{4} + 6$$9$$1$$12$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
$5$5.5.9.3$x^{5} + 80$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.3$x^{5} + 80$$5$$1$$9$$F_5$$[9/4]_{4}$