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Properties

Label 20.0.14567152137...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{6}\cdot 5^{15}\cdot 23^{6}\cdot 89^{7}$
Root discriminant $57.30$
Ramified primes $3, 5, 23, 89$
Class number $4624$ (GRH)
Class group $[2, 2, 1156]$ (GRH)
Galois group 20T466

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5729201, -17683985, 27457237, -24949939, 19281736, -12834474, 7673064, -4505558, 2527476, -1184443, 553045, -217497, 76643, -27411, 8136, -1799, 769, -37, 44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 44*x^18 - 37*x^17 + 769*x^16 - 1799*x^15 + 8136*x^14 - 27411*x^13 + 76643*x^12 - 217497*x^11 + 553045*x^10 - 1184443*x^9 + 2527476*x^8 - 4505558*x^7 + 7673064*x^6 - 12834474*x^5 + 19281736*x^4 - 24949939*x^3 + 27457237*x^2 - 17683985*x + 5729201)
 
gp: K = bnfinit(x^20 + 44*x^18 - 37*x^17 + 769*x^16 - 1799*x^15 + 8136*x^14 - 27411*x^13 + 76643*x^12 - 217497*x^11 + 553045*x^10 - 1184443*x^9 + 2527476*x^8 - 4505558*x^7 + 7673064*x^6 - 12834474*x^5 + 19281736*x^4 - 24949939*x^3 + 27457237*x^2 - 17683985*x + 5729201, 1)
 

Normalized defining polynomial

\( x^{20} + 44 x^{18} - 37 x^{17} + 769 x^{16} - 1799 x^{15} + 8136 x^{14} - 27411 x^{13} + 76643 x^{12} - 217497 x^{11} + 553045 x^{10} - 1184443 x^{9} + 2527476 x^{8} - 4505558 x^{7} + 7673064 x^{6} - 12834474 x^{5} + 19281736 x^{4} - 24949939 x^{3} + 27457237 x^{2} - 17683985 x + 5729201 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(145671521378968039543605010986328125=3^{6}\cdot 5^{15}\cdot 23^{6}\cdot 89^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.30$
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:  $3, 5, 23, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{5856680640395023908205540972837553051312728126817896719794318834894899} a^{19} + \frac{133189820786561001453062548418185419286219638011215522355637736986784}{5856680640395023908205540972837553051312728126817896719794318834894899} a^{18} + \frac{247319958792139138257217362168914233858977946333634439459336733416656}{5856680640395023908205540972837553051312728126817896719794318834894899} a^{17} + \frac{2588612033005735712694180115898728540171351218322471446125442430938560}{5856680640395023908205540972837553051312728126817896719794318834894899} a^{16} - \frac{1889289619801055079612376821428093540317436033391347046821272892250324}{5856680640395023908205540972837553051312728126817896719794318834894899} a^{15} - \frac{148140072634987223764752421154824555897945381484234116404972255934230}{5856680640395023908205540972837553051312728126817896719794318834894899} a^{14} - \frac{350821134467336180815480001186170595869560422343577606533205674275317}{5856680640395023908205540972837553051312728126817896719794318834894899} a^{13} - \frac{479716630312961410329177202541767950634142666340117287160346304009992}{5856680640395023908205540972837553051312728126817896719794318834894899} a^{12} + \frac{487117206966334920657351390043728181367028644939548551733664680535813}{5856680640395023908205540972837553051312728126817896719794318834894899} a^{11} + \frac{294738456057772447757857792999687639554232073673666311853727133142120}{5856680640395023908205540972837553051312728126817896719794318834894899} a^{10} - \frac{1219481191279908760209860490710622264058549437655107225978616351849994}{5856680640395023908205540972837553051312728126817896719794318834894899} a^{9} - \frac{1557329682267822386695159931631978857578026134528427127378464408655726}{5856680640395023908205540972837553051312728126817896719794318834894899} a^{8} - \frac{1616310724358589916007953901372575052167544927542986791282442897775533}{5856680640395023908205540972837553051312728126817896719794318834894899} a^{7} - \frac{1169795685909783271415252305794893799303657129030702094820934842618841}{5856680640395023908205540972837553051312728126817896719794318834894899} a^{6} - \frac{266827560746375790169653391316705446165206497420259484292005050530475}{5856680640395023908205540972837553051312728126817896719794318834894899} a^{5} + \frac{66147705443336241797498693852862357670036744303275982163466450424623}{5856680640395023908205540972837553051312728126817896719794318834894899} a^{4} - \frac{1503175880361443030147603956732014700854997762350667415284943828575173}{5856680640395023908205540972837553051312728126817896719794318834894899} a^{3} + \frac{461274982996937196093669217355394295608883101851450836069177080658184}{5856680640395023908205540972837553051312728126817896719794318834894899} a^{2} - \frac{406068354584435030838011265712872343886795298079002001110910115942608}{5856680640395023908205540972837553051312728126817896719794318834894899} a - \frac{2409443806558262945557858262399763658123049567875580538056980379080917}{5856680640395023908205540972837553051312728126817896719794318834894899}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{1156}$, which has order $4624$ (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:  $9$
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:  \( 366014.001413 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.767625.1, 10.10.2946240703125.1

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 $20$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.12.11.2$x^{12} - 20$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.8.6.2$x^{8} - 1633 x^{4} + 1270129$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
89Data not computed