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Properties

Label 20.12.1139285723...0000.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{17}\cdot 11^{4}\cdot 9931^{4}$
Root discriminant $79.95$
Ramified primes $2, 5, 11, 9931$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1010

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![45125, 0, -196750, 0, 185850, 0, -12250, 0, -43980, 0, 9205, 0, 3495, 0, -690, 0, -115, 0, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 10*x^18 - 115*x^16 - 690*x^14 + 3495*x^12 + 9205*x^10 - 43980*x^8 - 12250*x^6 + 185850*x^4 - 196750*x^2 + 45125)
 
gp: K = bnfinit(x^20 + 10*x^18 - 115*x^16 - 690*x^14 + 3495*x^12 + 9205*x^10 - 43980*x^8 - 12250*x^6 + 185850*x^4 - 196750*x^2 + 45125, 1)
 

Normalized defining polynomial

\( x^{20} + 10 x^{18} - 115 x^{16} - 690 x^{14} + 3495 x^{12} + 9205 x^{10} - 43980 x^{8} - 12250 x^{6} + 185850 x^{4} - 196750 x^{2} + 45125 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(113928572339566846848800000000000000000=2^{20}\cdot 5^{17}\cdot 11^{4}\cdot 9931^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.95$
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:  $2, 5, 11, 9931$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12}$, $\frac{1}{5} a^{13}$, $\frac{1}{5} a^{14}$, $\frac{1}{5} a^{15}$, $\frac{1}{25} a^{16} + \frac{2}{5} a^{10} - \frac{1}{5} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{4}$, $\frac{1}{25} a^{17} + \frac{2}{5} a^{11} - \frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{5}$, $\frac{1}{6425962832224864625} a^{18} + \frac{14937161896047679}{1285192566444972925} a^{16} - \frac{55742249820491896}{1285192566444972925} a^{14} + \frac{91366546673820552}{1285192566444972925} a^{12} - \frac{135561574773268316}{1285192566444972925} a^{10} + \frac{98462001110063176}{1285192566444972925} a^{8} - \frac{578052625430506621}{1285192566444972925} a^{6} - \frac{8205417701792889}{51407702657798917} a^{4} + \frac{12090100847261572}{257038513288994585} a^{2} - \frac{25526715824307640}{51407702657798917}$, $\frac{1}{122093293812272427875} a^{19} + \frac{33832053973888886}{4883731752490897115} a^{17} + \frac{2000565856491464784}{24418658762454485575} a^{15} + \frac{2404713166274771817}{24418658762454485575} a^{13} + \frac{121476938515726269}{24418658762454485575} a^{11} + \frac{12179272125692808671}{24418658762454485575} a^{9} - \frac{11373670183568279191}{24418658762454485575} a^{7} - \frac{452288709771355781}{4883731752490897115} a^{5} + \frac{12090100847261572}{4883731752490897115} a^{3} - \frac{282565229113302225}{976746350498179423} a$

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:  $15$
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:  \( 917804945731 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1010:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3686400
The 180 conjugacy class representatives for t20n1010 are not computed
Character table for t20n1010 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.932312193828125.1

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

Sibling fields

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 $20$ R $20$ R $20$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
5Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
9931Data not computed