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Properties

Label 10.2.51367706367...0000.1
Degree $10$
Signature $[2, 4]$
Discriminant $2^{8}\cdot 5^{11}\cdot 11^{8}\cdot 61^{8}$
Root discriminant $1866.68$
Ramified primes $2, 5, 11, 61$
Class number $23765625$ (GRH)
Class group $[5, 5, 5, 5, 195, 195]$ (GRH)
Galois group $F_5$ (as 10T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3190497936581, 62517065, -71448085, 53586070, -8930995, 3572393, -15, 10, 5, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 5*x^9 + 5*x^8 + 10*x^7 - 15*x^6 + 3572393*x^5 - 8930995*x^4 + 53586070*x^3 - 71448085*x^2 + 62517065*x + 3190497936581)
 
gp: K = bnfinit(x^10 - 5*x^9 + 5*x^8 + 10*x^7 - 15*x^6 + 3572393*x^5 - 8930995*x^4 + 53586070*x^3 - 71448085*x^2 + 62517065*x + 3190497936581, 1)
 

Normalized defining polynomial

\( x^{10} - 5 x^{9} + 5 x^{8} + 10 x^{7} - 15 x^{6} + 3572393 x^{5} - 8930995 x^{4} + 53586070 x^{3} - 71448085 x^{2} + 62517065 x + 3190497936581 \)

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:  \(513677063670173890032012500000000=2^{8}\cdot 5^{11}\cdot 11^{8}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1866.68$
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:  $2, 5, 11, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{11} a^{4} - \frac{2}{11} a^{3} - \frac{1}{11} a^{2} + \frac{2}{11} a + \frac{1}{11}$, $\frac{1}{55} a^{5} - \frac{1}{11} a^{3} + \frac{1}{11} a + \frac{24}{55}$, $\frac{1}{605} a^{6} - \frac{3}{605} a^{5} + \frac{1}{121} a^{3} + \frac{239}{605} a - \frac{122}{605}$, $\frac{1}{605} a^{7} + \frac{2}{605} a^{5} + \frac{1}{121} a^{4} - \frac{8}{121} a^{3} + \frac{239}{605} a^{2} + \frac{9}{121} a - \frac{102}{605}$, $\frac{1}{33275} a^{8} + \frac{7}{33275} a^{7} + \frac{13}{33275} a^{6} + \frac{239}{33275} a^{5} + \frac{2}{1331} a^{4} + \frac{14369}{33275} a^{3} - \frac{8017}{33275} a^{2} - \frac{10303}{33275} a - \frac{8634}{33275}$, $\frac{1}{338719616887699399409504964275} a^{9} + \frac{3379561614958960270863517}{338719616887699399409504964275} a^{8} - \frac{204853996716514274456394022}{338719616887699399409504964275} a^{7} + \frac{203458493187431391547250099}{338719616887699399409504964275} a^{6} - \frac{29790928195313707066938294}{13548784675507975976380198571} a^{5} - \frac{13507419285507620160601576356}{338719616887699399409504964275} a^{4} - \frac{5405359957903386473940802557}{30792692444336309037227724025} a^{3} - \frac{4769786671448611265176249593}{338719616887699399409504964275} a^{2} - \frac{120612241746918199885199854369}{338719616887699399409504964275} a - \frac{2424694577278949911968094765}{13548784675507975976380198571}$

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

Class group and class number

$C_{5}\times C_{5}\times C_{5}\times C_{5}\times C_{195}\times C_{195}$, which has order $23765625$ (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:  \( 140417.5079969902 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 10T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.10135847904050000.3

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 5 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$61$61.5.4.2$x^{5} + 122$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.4.2$x^{5} + 122$$5$$1$$4$$C_5$$[\ ]_{5}$