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Properties

Label 20.10.1288802115...0000.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{20}\cdot 5^{15}\cdot 3319^{5}$
Root discriminant $50.76$
Ramified primes $2, 5, 3319$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10371875, 0, 0, 0, 7807500, 0, -688125, 0, -132500, 0, -194875, 0, 40300, 0, -2925, 0, 300, 0, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 30*x^18 + 300*x^16 - 2925*x^14 + 40300*x^12 - 194875*x^10 - 132500*x^8 - 688125*x^6 + 7807500*x^4 - 10371875)
 
gp: K = bnfinit(x^20 - 30*x^18 + 300*x^16 - 2925*x^14 + 40300*x^12 - 194875*x^10 - 132500*x^8 - 688125*x^6 + 7807500*x^4 - 10371875, 1)
 

Normalized defining polynomial

\( x^{20} - 30 x^{18} + 300 x^{16} - 2925 x^{14} + 40300 x^{12} - 194875 x^{10} - 132500 x^{8} - 688125 x^{6} + 7807500 x^{4} - 10371875 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-12888021157371923168000000000000000=-\,2^{20}\cdot 5^{15}\cdot 3319^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.76$
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:  $2, 5, 3319$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{125} a^{14}$, $\frac{1}{125} a^{15}$, $\frac{1}{625} a^{16}$, $\frac{1}{625} a^{17}$, $\frac{1}{469260803308002297673363125} a^{18} + \frac{15006437826259189180396}{18770432132320091906934525} a^{16} - \frac{238378415985776119968619}{93852160661600459534672625} a^{14} + \frac{144637706327445162532187}{93852160661600459534672625} a^{12} - \frac{158865270486607842851017}{18770432132320091906934525} a^{10} - \frac{180584108731705853465996}{18770432132320091906934525} a^{8} - \frac{29812723179749920944557}{3754086426464018381386905} a^{6} - \frac{43369733673432000802241}{3754086426464018381386905} a^{4} - \frac{74365139359525972251784}{750817285292803676277381} a^{2} - \frac{22284097933280116987582}{750817285292803676277381}$, $\frac{1}{469260803308002297673363125} a^{19} + \frac{15006437826259189180396}{18770432132320091906934525} a^{17} - \frac{238378415985776119968619}{93852160661600459534672625} a^{15} + \frac{144637706327445162532187}{93852160661600459534672625} a^{13} - \frac{158865270486607842851017}{18770432132320091906934525} a^{11} - \frac{180584108731705853465996}{18770432132320091906934525} a^{9} - \frac{29812723179749920944557}{3754086426464018381386905} a^{7} - \frac{43369733673432000802241}{3754086426464018381386905} a^{5} - \frac{74365139359525972251784}{750817285292803676277381} a^{3} - \frac{22284097933280116987582}{750817285292803676277381} a$

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:  $14$
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:  \( 3902032155.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.34424253125.1

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

Sibling fields

Degree 20 siblings: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
5Data not computed
3319Data not computed