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Properties

Label 10.2.45166973116...0000.1
Degree $10$
Signature $[2, 4]$
Discriminant $2^{8}\cdot 5^{7}\cdot 7^{8}\cdot 23^{8}\cdot 29^{8}$
Root discriminant $4628.95$
Ramified primes $2, 5, 7, 23, 29$
Class number $75690000$ (GRH)
Class group $[5, 5, 1740, 1740]$ (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![-1750570287411125, 0, 12272470989561, 0, -14847792026, 0, 3466266, 0, -1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^8 + 3466266*x^6 - 14847792026*x^4 + 12272470989561*x^2 - 1750570287411125)
 
gp: K = bnfinit(x^10 - x^8 + 3466266*x^6 - 14847792026*x^4 + 12272470989561*x^2 - 1750570287411125, 1)
 

Normalized defining polynomial

\( x^{10} - x^{8} + 3466266 x^{6} - 14847792026 x^{4} + 12272470989561 x^{2} - 1750570287411125 \)

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:  \(4516697311642659815261691676820000000=2^{8}\cdot 5^{7}\cdot 7^{8}\cdot 23^{8}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $4628.95$
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:  $2, 5, 7, 23, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{29} a^{4} - \frac{12}{29} a^{2} + \frac{7}{29}$, $\frac{1}{58} a^{5} - \frac{1}{58} a^{4} - \frac{6}{29} a^{3} + \frac{6}{29} a^{2} + \frac{7}{58} a - \frac{7}{58}$, $\frac{1}{184788} a^{6} - \frac{1}{61596} a^{4} - \frac{1}{2} a^{3} + \frac{38179}{184788} a^{2} + \frac{28889}{184788}$, $\frac{1}{184788} a^{7} - \frac{1}{61596} a^{5} - \frac{1}{58} a^{4} + \frac{38179}{184788} a^{3} + \frac{6}{29} a^{2} + \frac{28889}{184788} a + \frac{11}{29}$, $\frac{1}{2422785920877612} a^{8} + \frac{352379255}{605696480219403} a^{6} - \frac{170915160616}{10266042037617} a^{4} - \frac{1}{2} a^{3} + \frac{453687747983291}{1211392960438806} a^{2} - \frac{1}{2} a + \frac{270801156621469}{2422785920877612}$, $\frac{1}{45333558998824492132020} a^{9} + \frac{1030721461887841}{11333389749706123033005} a^{7} + \frac{57741336537267898763}{22666779499412246066010} a^{5} - \frac{3648928518936339131881}{7555593166470748688670} a^{3} - \frac{1}{2} a^{2} - \frac{6937593187530213614749}{45333558998824492132020} a - \frac{1}{2}$

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

Class group and class number

$C_{5}\times C_{5}\times C_{1740}\times C_{1740}$, which has order $75690000$ (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:  \( 5542985.379769533 \) (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

\(\Q(\sqrt{5}) \), 5.1.950441719585442000.1

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 R ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R R ${\href{/LocalNumberField/31.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$[\ ]_{5}^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$7$7.10.8.1$x^{10} - 7 x^{5} + 147$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$23$23.10.8.1$x^{10} - 23 x^{5} + 3703$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$29$29.5.4.1$x^{5} - 29$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
29.5.4.1$x^{5} - 29$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$