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Properties

Label 10.2.63613148112...0000.1
Degree $10$
Signature $[2, 4]$
Discriminant $2^{8}\cdot 3^{9}\cdot 5^{18}\cdot 13^{9}\cdot 199^{5}$
Root discriminant $12{,}032.47$
Ramified primes $2, 3, 5, 13, 199$
Class number $146173440$ (GRH)
Class group $[2, 2, 4, 4, 2283960]$ (GRH)
Galois group $F_{5}\times C_2$ (as 10T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27481198710005660, -67385222095700, 70745098715400, 149531844600, -72901398000, -112686681, 37577805, 38790, -9690, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 5*x^9 - 9690*x^8 + 38790*x^7 + 37577805*x^6 - 112686681*x^5 - 72901398000*x^4 + 149531844600*x^3 + 70745098715400*x^2 - 67385222095700*x - 27481198710005660)
 
gp: K = bnfinit(x^10 - 5*x^9 - 9690*x^8 + 38790*x^7 + 37577805*x^6 - 112686681*x^5 - 72901398000*x^4 + 149531844600*x^3 + 70745098715400*x^2 - 67385222095700*x - 27481198710005660, 1)
 

Normalized defining polynomial

\( x^{10} - 5 x^{9} - 9690 x^{8} + 38790 x^{7} + 37577805 x^{6} - 112686681 x^{5} - 72901398000 x^{4} + 149531844600 x^{3} + 70745098715400 x^{2} - 67385222095700 x - 27481198710005660 \)

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:  \(63613148112891287123047110351562500000000=2^{8}\cdot 3^{9}\cdot 5^{18}\cdot 13^{9}\cdot 199^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12{,}032.47$
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:  $2, 3, 5, 13, 199$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{39} a^{3} + \frac{5}{39} a^{2} + \frac{4}{39} a - \frac{6}{13}$, $\frac{1}{117} a^{4} + \frac{1}{117} a^{3} - \frac{1}{39} a^{2} + \frac{31}{117} a + \frac{7}{117}$, $\frac{1}{1521} a^{5} + \frac{4}{1521} a^{4} + \frac{1}{169} a^{3} + \frac{106}{1521} a^{2} - \frac{20}{1521} a + \frac{200}{507}$, $\frac{1}{9126} a^{6} - \frac{1}{3042} a^{5} + \frac{11}{3042} a^{4} - \frac{61}{9126} a^{3} + \frac{55}{1521} a^{2} - \frac{50}{1521} a + \frac{500}{4563}$, $\frac{1}{9126} a^{7} + \frac{10}{4563} a^{4} + \frac{1}{1014} a^{3} + \frac{53}{507} a^{2} - \frac{22}{4563} a + \frac{73}{507}$, $\frac{1}{355914} a^{8} - \frac{2}{177957} a^{7} + \frac{7}{355914} a^{6} - \frac{7}{355914} a^{5} - \frac{283}{177957} a^{4} + \frac{1139}{355914} a^{3} - \frac{4135}{177957} a^{2} + \frac{3850}{177957} a - \frac{14500}{177957}$, $\frac{1}{5858178767928564610701778517466} a^{9} - \frac{2424399305877956053237271}{5858178767928564610701778517466} a^{8} + \frac{89402938707881015604926278}{2929089383964282305350889258733} a^{7} - \frac{40213345695241499277812881}{2929089383964282305350889258733} a^{6} + \frac{57056083366226339939400209}{450629135994504970053982962882} a^{5} - \frac{11941511496483330956185011395}{5858178767928564610701778517466} a^{4} + \frac{14609150071633969714372855093}{2929089383964282305350889258733} a^{3} - \frac{365354504670247549379156776786}{2929089383964282305350889258733} a^{2} - \frac{962147883662959415830591966517}{2929089383964282305350889258733} a - \frac{490857257717102009131146954395}{2929089383964282305350889258733}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{2283960}$, which has order $146173440$ (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:  \( 1276041451.9892254 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 10T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $F_{5}\times C_2$
Character table for $F_{5}\times C_2$

Intermediate fields

\(\Q(\sqrt{7761}) \), 5.1.72295031250000.26

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

Sibling fields

Galois closure: data not computed
Degree 10 sibling: data not computed
Degree 20 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$3$3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.5.9.2$x^{5} + 55$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.2$x^{5} + 55$$5$$1$$9$$F_5$$[9/4]_{4}$
$13$13.10.9.1$x^{10} - 13$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$199$199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.4.2.1$x^{4} + 2189 x^{2} + 1425636$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
199.4.2.1$x^{4} + 2189 x^{2} + 1425636$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$