↑

Properties

Label 20.20.3108100390...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{56}\cdot 3^{28}\cdot 5^{8}\cdot 13^{6}$
Root discriminant $133.24$
Ramified primes $2, 3, 5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T201

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7056, 0, -3857880, 0, 22734657, 0, -38331864, 0, 22526100, 0, -6252552, 0, 930918, 0, -78696, 0, 3780, 0, -96, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 96*x^18 + 3780*x^16 - 78696*x^14 + 930918*x^12 - 6252552*x^10 + 22526100*x^8 - 38331864*x^6 + 22734657*x^4 - 3857880*x^2 + 7056)
 
gp: K = bnfinit(x^20 - 96*x^18 + 3780*x^16 - 78696*x^14 + 930918*x^12 - 6252552*x^10 + 22526100*x^8 - 38331864*x^6 + 22734657*x^4 - 3857880*x^2 + 7056, 1)
 

Normalized defining polynomial

\( x^{20} - 96 x^{18} + 3780 x^{16} - 78696 x^{14} + 930918 x^{12} - 6252552 x^{10} + 22526100 x^{8} - 38331864 x^{6} + 22734657 x^{4} - 3857880 x^{2} + 7056 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3108100390178909231545231736202854400000000=2^{56}\cdot 3^{28}\cdot 5^{8}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $133.24$
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:  $2, 3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{3}{8} a^{4} + \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{160} a^{10} - \frac{1}{20} a^{8} + \frac{9}{80} a^{6} + \frac{9}{40} a^{4} - \frac{3}{32} a^{2} - \frac{11}{40}$, $\frac{1}{320} a^{11} - \frac{1}{40} a^{9} + \frac{9}{160} a^{7} - \frac{1}{8} a^{6} + \frac{9}{80} a^{5} + \frac{1}{4} a^{4} - \frac{3}{64} a^{3} + \frac{1}{8} a^{2} - \frac{11}{80} a - \frac{1}{2}$, $\frac{1}{320} a^{12} - \frac{3}{160} a^{8} + \frac{1}{16} a^{6} - \frac{87}{320} a^{4} - \frac{1}{2} a^{3} - \frac{21}{80} a^{2} - \frac{1}{2} a - \frac{1}{10}$, $\frac{1}{640} a^{13} - \frac{1}{640} a^{11} - \frac{1}{320} a^{10} + \frac{1}{320} a^{9} + \frac{1}{40} a^{8} - \frac{19}{320} a^{7} + \frac{11}{160} a^{6} + \frac{117}{640} a^{5} + \frac{11}{80} a^{4} - \frac{189}{640} a^{3} - \frac{5}{64} a^{2} + \frac{43}{160} a - \frac{29}{80}$, $\frac{1}{640} a^{14} - \frac{1}{640} a^{12} - \frac{1}{320} a^{10} - \frac{3}{320} a^{8} + \frac{9}{128} a^{6} - \frac{13}{640} a^{4} - \frac{1}{2} a^{3} - \frac{11}{80} a^{2} + \frac{11}{40}$, $\frac{1}{640} a^{15} - \frac{1}{640} a^{11} - \frac{1}{320} a^{10} - \frac{1}{32} a^{9} + \frac{1}{40} a^{8} - \frac{37}{640} a^{7} + \frac{11}{160} a^{6} + \frac{1}{40} a^{5} + \frac{11}{80} a^{4} + \frac{93}{640} a^{3} + \frac{27}{64} a^{2} - \frac{3}{32} a - \frac{29}{80}$, $\frac{1}{2560} a^{16} - \frac{1}{1280} a^{14} + \frac{1}{2560} a^{12} + \frac{1}{640} a^{10} - \frac{1}{16} a^{9} + \frac{11}{512} a^{8} - \frac{1}{16} a^{7} + \frac{103}{1280} a^{6} - \frac{1}{16} a^{5} - \frac{1161}{2560} a^{4} + \frac{7}{16} a^{3} - \frac{153}{320} a^{2} + \frac{1}{4} a - \frac{17}{160}$, $\frac{1}{2560} a^{17} - \frac{1}{1280} a^{15} + \frac{1}{2560} a^{13} - \frac{1}{640} a^{11} + \frac{119}{2560} a^{9} - \frac{1}{16} a^{8} + \frac{31}{1280} a^{7} - \frac{1}{16} a^{6} - \frac{169}{2560} a^{5} + \frac{7}{16} a^{4} - \frac{69}{160} a^{3} + \frac{7}{16} a^{2} - \frac{15}{32} a - \frac{1}{4}$, $\frac{1}{35375733662737920} a^{18} + \frac{248329278401}{2947977805228160} a^{16} - \frac{1497128531429}{11791911220912640} a^{14} - \frac{8552876060387}{5895955610456320} a^{12} - \frac{29245299656947}{11791911220912640} a^{10} - \frac{1}{16} a^{9} + \frac{70318571544033}{1473988902614080} a^{8} - \frac{1}{16} a^{7} + \frac{828885758866109}{11791911220912640} a^{6} - \frac{1}{16} a^{5} + \frac{369708169093229}{1179191122091264} a^{4} - \frac{1}{16} a^{3} - \frac{7397744971781}{736994451307040} a^{2} + \frac{1}{4} a - \frac{28290575330469}{73699445130704}$, $\frac{1}{495260271278330880} a^{19} - \frac{1}{70751467325475840} a^{18} + \frac{24024393716949}{165086757092776960} a^{17} + \frac{722579641413}{4716764488365056} a^{16} + \frac{3378984851599}{4716764488365056} a^{15} + \frac{10709559172767}{23583822441825280} a^{14} - \frac{123048704496161}{165086757092776960} a^{13} + \frac{3287106158767}{23583822441825280} a^{12} - \frac{231918773766383}{165086757092776960} a^{11} + \frac{10820438374271}{23583822441825280} a^{10} + \frac{4740385868199047}{165086757092776960} a^{9} + \frac{210846601040511}{4716764488365056} a^{8} - \frac{60848642003693}{33017351418555392} a^{7} + \frac{484620451505241}{4716764488365056} a^{6} + \frac{9772679698894701}{165086757092776960} a^{5} + \frac{3981479248622933}{23583822441825280} a^{4} + \frac{1881150536502509}{10317922318298560} a^{3} - \frac{708380315401471}{2947977805228160} a^{2} + \frac{1711585480544987}{10317922318298560} a + \frac{38776341309233}{1473988902614080}$

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:  $19$
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:  \( 14681979145400000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T201:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1440
The 13 conjugacy class representatives for t20n201
Character table for t20n201

Intermediate fields

\(\Q(\sqrt{3}) \), 10.10.293830090272276480000.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 12 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 sibling: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.8.24.77$x^{8} - 3$$8$$1$$24$$Z_8 : Z_8^\times$$[2, 2, 3, 4]^{2}$
2.8.24.77$x^{8} - 3$$8$$1$$24$$Z_8 : Z_8^\times$$[2, 2, 3, 4]^{2}$
3Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$