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Properties

Label 20.0.43542784730...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{12}\cdot 257^{4}\cdot 431^{4}\cdot 1031^{4}$
Root discriminant $214.76$
Ramified primes $2, 5, 257, 431, 1031$
Class number $203718208$ (GRH)
Class group $[2, 4, 25464776]$ (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![34507149121, 0, 347793632904, 0, 342449474695, 0, 123475672018, 0, 19617373055, 0, 1637220249, 0, 77860477, 0, 2164086, 0, 34523, 0, 291, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 291*x^18 + 34523*x^16 + 2164086*x^14 + 77860477*x^12 + 1637220249*x^10 + 19617373055*x^8 + 123475672018*x^6 + 342449474695*x^4 + 347793632904*x^2 + 34507149121)
 
gp: K = bnfinit(x^20 + 291*x^18 + 34523*x^16 + 2164086*x^14 + 77860477*x^12 + 1637220249*x^10 + 19617373055*x^8 + 123475672018*x^6 + 342449474695*x^4 + 347793632904*x^2 + 34507149121, 1)
 

Normalized defining polynomial

\( x^{20} + 291 x^{18} + 34523 x^{16} + 2164086 x^{14} + 77860477 x^{12} + 1637220249 x^{10} + 19617373055 x^{8} + 123475672018 x^{6} + 342449474695 x^{4} + 347793632904 x^{2} + 34507149121 \)

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:  \(43542784730387716079982562074992896000000000000=2^{20}\cdot 5^{12}\cdot 257^{4}\cdot 431^{4}\cdot 1031^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $214.76$
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:  $2, 5, 257, 431, 1031$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1293} a^{14} + \frac{97}{431} a^{12} - \frac{388}{1293} a^{10} + \frac{466}{1293} a^{8} - \frac{104}{1293} a^{6} - \frac{487}{1293} a^{4} - \frac{34}{431} a^{2} + \frac{1}{3}$, $\frac{1}{1293} a^{15} + \frac{97}{431} a^{13} - \frac{388}{1293} a^{11} + \frac{466}{1293} a^{9} - \frac{104}{1293} a^{7} - \frac{487}{1293} a^{5} - \frac{34}{431} a^{3} + \frac{1}{3} a$, $\frac{1}{1671849} a^{16} - \frac{571}{1671849} a^{14} + \frac{340964}{1671849} a^{12} - \frac{102112}{1671849} a^{10} + \frac{184577}{557283} a^{8} - \frac{730601}{1671849} a^{6} - \frac{54839}{1671849} a^{4} - \frac{311}{3879} a^{2} + \frac{1}{9}$, $\frac{1}{1671849} a^{17} - \frac{571}{1671849} a^{15} + \frac{340964}{1671849} a^{13} - \frac{102112}{1671849} a^{11} + \frac{184577}{557283} a^{9} - \frac{730601}{1671849} a^{7} - \frac{54839}{1671849} a^{5} - \frac{311}{3879} a^{3} + \frac{1}{9} a$, $\frac{1}{1021998928816701098571305744251898845742046} a^{18} + \frac{111813741257832915539856983992893113}{510999464408350549285652872125949422871023} a^{16} - \frac{26028926321328971373337982234019633895}{113555436535189010952367304916877649526894} a^{14} - \frac{38822303207840762202073350166048843429165}{113555436535189010952367304916877649526894} a^{12} - \frac{71216813313556667159010096094008360653767}{510999464408350549285652872125949422871023} a^{10} - \frac{416686687079643723403723963013886936706817}{1021998928816701098571305744251898845742046} a^{8} - \frac{3582695458621968441700187739338595065602}{56777718267594505476183652458438824763447} a^{6} - \frac{585597668135920123560411992060625302426}{1185613606515894545906387174306147152833} a^{4} + \frac{2535792609657839225481048301312646747}{5501687269215287916038919602348710686} a^{2} + \frac{2025652718666053142017737092543635}{12764935659432222543013734576215106}$, $\frac{1}{1021998928816701098571305744251898845742046} a^{19} + \frac{111813741257832915539856983992893113}{510999464408350549285652872125949422871023} a^{17} - \frac{26028926321328971373337982234019633895}{113555436535189010952367304916877649526894} a^{15} - \frac{38822303207840762202073350166048843429165}{113555436535189010952367304916877649526894} a^{13} - \frac{71216813313556667159010096094008360653767}{510999464408350549285652872125949422871023} a^{11} - \frac{416686687079643723403723963013886936706817}{1021998928816701098571305744251898845742046} a^{9} - \frac{3582695458621968441700187739338595065602}{56777718267594505476183652458438824763447} a^{7} - \frac{585597668135920123560411992060625302426}{1185613606515894545906387174306147152833} a^{5} + \frac{2535792609657839225481048301312646747}{5501687269215287916038919602348710686} a^{3} + \frac{2025652718666053142017737092543635}{12764935659432222543013734576215106} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{25464776}$, which has order $203718208$ (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:  \( 12637712.6049 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.1096992360765625.1

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

Sibling fields

Degree 20 sibling: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $16{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
257Data not computed
431Data not computed
1031Data not computed