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Properties

Label 20.8.64303034622...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 5^{13}\cdot 3469^{5}$
Root discriminant $43.69$
Ramified primes $2, 5, 3469$
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![431, 3188, 2348, -16766, -13423, 1680, 20642, 12234, -2302, 5342, -9710, 932, 3258, -2052, 34, 258, -129, 22, 10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 10*x^18 + 22*x^17 - 129*x^16 + 258*x^15 + 34*x^14 - 2052*x^13 + 3258*x^12 + 932*x^11 - 9710*x^10 + 5342*x^9 - 2302*x^8 + 12234*x^7 + 20642*x^6 + 1680*x^5 - 13423*x^4 - 16766*x^3 + 2348*x^2 + 3188*x + 431)
 
gp: K = bnfinit(x^20 - 2*x^19 + 10*x^18 + 22*x^17 - 129*x^16 + 258*x^15 + 34*x^14 - 2052*x^13 + 3258*x^12 + 932*x^11 - 9710*x^10 + 5342*x^9 - 2302*x^8 + 12234*x^7 + 20642*x^6 + 1680*x^5 - 13423*x^4 - 16766*x^3 + 2348*x^2 + 3188*x + 431, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 10 x^{18} + 22 x^{17} - 129 x^{16} + 258 x^{15} + 34 x^{14} - 2052 x^{13} + 3258 x^{12} + 932 x^{11} - 9710 x^{10} + 5342 x^{9} - 2302 x^{8} + 12234 x^{7} + 20642 x^{6} + 1680 x^{5} - 13423 x^{4} - 16766 x^{3} + 2348 x^{2} + 3188 x + 431 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(643030346221425086720000000000000=2^{20}\cdot 5^{13}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.69$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{2}$, $\frac{1}{5345038174514814329987380413296242770721434238} a^{19} - \frac{322728176685832871497836089660713059995813204}{2672519087257407164993690206648121385360717119} a^{18} - \frac{991445035163773219512225561187246538822321121}{5345038174514814329987380413296242770721434238} a^{17} - \frac{21145417849705118823468691824980092483462931}{2672519087257407164993690206648121385360717119} a^{16} + \frac{46711515217564683990002236032487558720530829}{5345038174514814329987380413296242770721434238} a^{15} + \frac{348051626758777268281146063661754874340380210}{2672519087257407164993690206648121385360717119} a^{14} + \frac{176459900269686928689054389483065970744719104}{2672519087257407164993690206648121385360717119} a^{13} - \frac{1039240747155171573884401850775218021055914095}{5345038174514814329987380413296242770721434238} a^{12} - \frac{634277762927568249401306992383623902641952783}{2672519087257407164993690206648121385360717119} a^{11} + \frac{616823285184270233657515757590004843540119662}{2672519087257407164993690206648121385360717119} a^{10} + \frac{1252625839099087405564387134081260281766754963}{5345038174514814329987380413296242770721434238} a^{9} - \frac{2492054148519475185245429497718501671407783239}{5345038174514814329987380413296242770721434238} a^{8} + \frac{1162771925869472714087868188720259715493760949}{5345038174514814329987380413296242770721434238} a^{7} - \frac{550179998406802985465632950436761304456677349}{2672519087257407164993690206648121385360717119} a^{6} + \frac{1270635406448218623735601323845278272086443298}{2672519087257407164993690206648121385360717119} a^{5} + \frac{485022038881017461198599607430507781801429387}{5345038174514814329987380413296242770721434238} a^{4} + \frac{2608465503696823022221794617106039509589683091}{5345038174514814329987380413296242770721434238} a^{3} - \frac{1055656112863349055399476794144160366315936818}{2672519087257407164993690206648121385360717119} a^{2} + \frac{914451176445509373363436848855255951743306651}{2672519087257407164993690206648121385360717119} a - \frac{5357120168634357826548933847218909527454349}{12401480683329035568416195854515644479632098}$

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:  $13$
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:  \( 897831739.909 \) (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

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
3469Data not computed