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Properties

Label 20.10.4223146772...0000.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{30}\cdot 5^{10}\cdot 3319^{5}$
Root discriminant $48.00$
Ramified primes $2, 5, 3319$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T756

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3398656, 0, 7089152, 0, -3959552, 0, 400128, 0, 81344, 0, 32512, 0, -6112, 0, -1232, 0, -96, 0, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 10*x^18 - 96*x^16 - 1232*x^14 - 6112*x^12 + 32512*x^10 + 81344*x^8 + 400128*x^6 - 3959552*x^4 + 7089152*x^2 - 3398656)
 
gp: K = bnfinit(x^20 + 10*x^18 - 96*x^16 - 1232*x^14 - 6112*x^12 + 32512*x^10 + 81344*x^8 + 400128*x^6 - 3959552*x^4 + 7089152*x^2 - 3398656, 1)
 

Normalized defining polynomial

\( x^{20} + 10 x^{18} - 96 x^{16} - 1232 x^{14} - 6112 x^{12} + 32512 x^{10} + 81344 x^{8} + 400128 x^{6} - 3959552 x^{4} + 7089152 x^{2} - 3398656 \)

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:  \(-4223146772847631783690240000000000=-\,2^{30}\cdot 5^{10}\cdot 3319^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.00$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{384} a^{14} - \frac{1}{192} a^{12} + \frac{1}{24} a^{6} - \frac{1}{12} a^{4} - \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{384} a^{15} - \frac{1}{192} a^{13} + \frac{1}{24} a^{7} - \frac{1}{12} a^{5} - \frac{1}{6} a^{3} + \frac{1}{3} a$, $\frac{1}{2304} a^{16} + \frac{1}{1152} a^{14} + \frac{1}{144} a^{12} - \frac{1}{96} a^{10} + \frac{1}{144} a^{8} - \frac{1}{36} a^{6} - \frac{1}{12} a^{4} - \frac{2}{9} a^{2} - \frac{4}{9}$, $\frac{1}{2304} a^{17} + \frac{1}{1152} a^{15} + \frac{1}{144} a^{13} - \frac{1}{96} a^{11} + \frac{1}{144} a^{9} - \frac{1}{36} a^{7} - \frac{1}{12} a^{5} - \frac{2}{9} a^{3} - \frac{4}{9} a$, $\frac{1}{138699985838169005568} a^{18} + \frac{1617461532902467}{23116664306361500928} a^{16} + \frac{3487026850587169}{5779166076590375232} a^{14} - \frac{56043304892932097}{8668749114885562848} a^{12} - \frac{3704786389752769}{270898409840173839} a^{10} + \frac{501671237569631}{160532391016399312} a^{8} - \frac{39101906714246981}{1083593639360695356} a^{6} + \frac{20296032505360285}{541796819680347678} a^{4} + \frac{6710665556489969}{180598939893449226} a^{2} - \frac{10542485044711511}{270898409840173839}$, $\frac{1}{138699985838169005568} a^{19} + \frac{1617461532902467}{23116664306361500928} a^{17} + \frac{3487026850587169}{5779166076590375232} a^{15} - \frac{56043304892932097}{8668749114885562848} a^{13} - \frac{3704786389752769}{270898409840173839} a^{11} + \frac{501671237569631}{160532391016399312} a^{9} - \frac{39101906714246981}{1083593639360695356} a^{7} + \frac{20296032505360285}{541796819680347678} a^{5} + \frac{6710665556489969}{180598939893449226} a^{3} - \frac{10542485044711511}{270898409840173839} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1274535897.73 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T756:

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 t20n756 are not computed
Character table for t20n756 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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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