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Properties

Label 10.2.65684083557...0625.2
Degree $10$
Signature $[2, 4]$
Discriminant $3^{16}\cdot 5^{16}$
Root discriminant $76.16$
Ramified primes $3, 5$
Class number $20$ (GRH)
Class group $[20]$ (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![37800, 19125, 12150, -3825, -975, -18, 60, -35, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 15*x^8 - 35*x^7 + 60*x^6 - 18*x^5 - 975*x^4 - 3825*x^3 + 12150*x^2 + 19125*x + 37800)
 
gp: K = bnfinit(x^10 - 15*x^8 - 35*x^7 + 60*x^6 - 18*x^5 - 975*x^4 - 3825*x^3 + 12150*x^2 + 19125*x + 37800, 1)
 

Normalized defining polynomial

\( x^{10} - 15 x^{8} - 35 x^{7} + 60 x^{6} - 18 x^{5} - 975 x^{4} - 3825 x^{3} + 12150 x^{2} + 19125 x + 37800 \)

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:  \(6568408355712890625=3^{16}\cdot 5^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.16$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3}$, $\frac{1}{27} a^{7} + \frac{1}{9} a^{5} + \frac{7}{27} a^{4} + \frac{2}{9} a^{3} - \frac{1}{3} a^{2} - \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{135} a^{8} - \frac{1}{9} a^{6} - \frac{4}{27} a^{5} + \frac{4}{9} a^{4} + \frac{1}{5} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{97055303366745} a^{9} - \frac{165067802054}{97055303366745} a^{8} - \frac{111872449699}{19411060673349} a^{7} - \frac{3177711475396}{19411060673349} a^{6} + \frac{3810797738735}{19411060673349} a^{5} - \frac{2565816367013}{97055303366745} a^{4} - \frac{3270721044662}{10783922596305} a^{3} + \frac{2043481107404}{6470353557783} a^{2} + \frac{665200340228}{6470353557783} a - \frac{285721402199}{2156784519261}$

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

Class group and class number

$C_{20}$, which has order $20$ (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:  \( 267506.739891 \) (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: data not computed
Degree 15 siblings: data not computed
Degree 20 sibling: data not computed
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.3.5.2$x^{3} + 21$$3$$1$$5$$S_3$$[5/2]_{2}$
3.6.11.7$x^{6} + 21$$6$$1$$11$$S_3$$[5/2]_{2}$
$5$5.5.8.1$x^{5} - 5 x^{4} + 105$$5$$1$$8$$C_5$$[2]$
5.5.8.1$x^{5} - 5 x^{4} + 105$$5$$1$$8$$C_5$$[2]$