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Properties

Label 10.2.94119200000...0000.1
Degree $10$
Signature $[2, 4]$
Discriminant $2^{22}\cdot 5^{19}\cdot 7^{6}$
Root discriminant $314.32$
Ramified primes $2, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5^2 : Q_8$ (as 10T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6182000, -225000, -39375, 255000, 52500, -29400, 1750, -600, 100, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 + 100*x^8 - 600*x^7 + 1750*x^6 - 29400*x^5 + 52500*x^4 + 255000*x^3 - 39375*x^2 - 225000*x - 6182000)
 
gp: K = bnfinit(x^10 + 100*x^8 - 600*x^7 + 1750*x^6 - 29400*x^5 + 52500*x^4 + 255000*x^3 - 39375*x^2 - 225000*x - 6182000, 1)
 

Normalized defining polynomial

\( x^{10} + 100 x^{8} - 600 x^{7} + 1750 x^{6} - 29400 x^{5} + 52500 x^{4} + 255000 x^{3} - 39375 x^{2} - 225000 x - 6182000 \)

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:  \(9411920000000000000000000=2^{22}\cdot 5^{19}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $314.32$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{20} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{2240} a^{5} + \frac{5}{224} a^{3} - \frac{15}{112} a^{2} - \frac{75}{448} a - \frac{41}{112}$, $\frac{1}{2240} a^{6} + \frac{5}{224} a^{4} - \frac{15}{112} a^{3} - \frac{75}{448} a^{2} - \frac{41}{112} a$, $\frac{1}{11200} a^{7} + \frac{13}{560} a^{4} - \frac{3}{448} a^{3} + \frac{13}{112} a^{2} - \frac{17}{224} a - \frac{19}{56}$, $\frac{1}{22400} a^{8} - \frac{1}{4480} a^{6} + \frac{47}{4480} a^{4} - \frac{9}{112} a^{3} - \frac{311}{896} a^{2} - \frac{29}{112} a + \frac{1}{56}$, $\frac{1}{25491200} a^{9} - \frac{257}{25491200} a^{8} - \frac{993}{25491200} a^{7} + \frac{393}{5098240} a^{6} + \frac{631}{5098240} a^{5} + \frac{11759}{728320} a^{4} - \frac{30333}{145664} a^{3} + \frac{467367}{1019648} a^{2} - \frac{57201}{127456} a + \frac{1137}{9104}$

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:  \( 811071557.87 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:Q_8$ (as 10T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 8 conjugacy class representatives for $C_5^2 : Q_8$
Character table for $C_5^2 : Q_8$

Intermediate fields

\(\Q(\sqrt{5}) \)

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 25 sibling: 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R R ${\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.22.3$x^{8} + 8 x^{6} + 6 x^{4} + 16 x^{2} + 16 x + 4$$4$$2$$22$$Q_8$$[3, 4]^{2}$
$5$5.10.19.8$x^{10} - 10 x^{5} + 105$$10$$1$$19$$C_5^2 : C_4$$[7/4, 9/4]_{4}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$