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Properties

Label 20.20.7285959626...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{40}\cdot 5^{10}\cdot 13^{4}\cdot 29^{4}\cdot 31^{4}\cdot 1381^{4}$
Root discriminant $247.24$
Ramified primes $2, 5, 13, 29, 31, 1381$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1013

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![931139348459776, 0, -433542795843328, 0, 82763463901760, 0, -8538377065024, 0, 528715968864, 0, -20646179760, 0, 517755720, 0, -8271656, 0, 80864, 0, -438, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 438*x^18 + 80864*x^16 - 8271656*x^14 + 517755720*x^12 - 20646179760*x^10 + 528715968864*x^8 - 8538377065024*x^6 + 82763463901760*x^4 - 433542795843328*x^2 + 931139348459776)
 
gp: K = bnfinit(x^20 - 438*x^18 + 80864*x^16 - 8271656*x^14 + 517755720*x^12 - 20646179760*x^10 + 528715968864*x^8 - 8538377065024*x^6 + 82763463901760*x^4 - 433542795843328*x^2 + 931139348459776, 1)
 

Normalized defining polynomial

\( x^{20} - 438 x^{18} + 80864 x^{16} - 8271656 x^{14} + 517755720 x^{12} - 20646179760 x^{10} + 528715968864 x^{8} - 8538377065024 x^{6} + 82763463901760 x^{4} - 433542795843328 x^{2} + 931139348459776 \)

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:  \(728595962694391038340952582792874557440000000000=2^{40}\cdot 5^{10}\cdot 13^{4}\cdot 29^{4}\cdot 31^{4}\cdot 1381^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $247.24$
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:  $2, 5, 13, 29, 31, 1381$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{4} a^{6}$, $\frac{1}{4} a^{7}$, $\frac{1}{8} a^{8}$, $\frac{1}{8} a^{9}$, $\frac{1}{8} a^{10}$, $\frac{1}{16} a^{11}$, $\frac{1}{16} a^{12}$, $\frac{1}{16} a^{13}$, $\frac{1}{44192} a^{14} - \frac{219}{22096} a^{12} - \frac{499}{11048} a^{10} - \frac{557}{11048} a^{8} + \frac{281}{5524} a^{6} - \frac{275}{1381} a^{4} + \frac{262}{1381} a^{2}$, $\frac{1}{44192} a^{15} - \frac{219}{22096} a^{13} + \frac{383}{22096} a^{11} - \frac{557}{11048} a^{9} + \frac{281}{5524} a^{7} - \frac{275}{1381} a^{5} + \frac{262}{1381} a^{3}$, $\frac{1}{122058304} a^{16} - \frac{219}{61029152} a^{14} + \frac{2527}{3814322} a^{12} + \frac{218301}{3814322} a^{10} - \frac{124009}{15257288} a^{8} + \frac{380881}{3814322} a^{6} + \frac{319142}{1907161} a^{4} - \frac{368}{1381} a^{2}$, $\frac{1}{122058304} a^{17} - \frac{219}{61029152} a^{15} + \frac{2527}{3814322} a^{13} - \frac{160753}{30514576} a^{11} - \frac{124009}{15257288} a^{9} + \frac{380881}{3814322} a^{7} + \frac{319142}{1907161} a^{5} + \frac{645}{2762} a^{3}$, $\frac{1}{14595744302166943289723868678589440623104} a^{18} + \frac{3413311921051811064818159809475}{3648936075541735822430967169647360155776} a^{16} + \frac{19444826825979720811931891895429009}{1824468037770867911215483584823680077888} a^{14} - \frac{27934410550923516233659371673031638975}{1824468037770867911215483584823680077888} a^{12} - \frac{50568809182262617613846861577634296733}{1824468037770867911215483584823680077888} a^{10} - \frac{6347059904524269377783386574992440195}{114029252360679244450967724051480004868} a^{8} - \frac{7573857719159697499639664608865460361}{456117009442716977803870896205920019472} a^{6} - \frac{12831914381586293689293078810953931}{82570059638435368900049039863490228} a^{4} + \frac{12467242672483215209089957442205}{119580100852187355394712584885576} a^{2} + \frac{6898427548967347129303269671}{43294750489568195291351406548}$, $\frac{1}{14595744302166943289723868678589440623104} a^{19} + \frac{3413311921051811064818159809475}{3648936075541735822430967169647360155776} a^{17} + \frac{19444826825979720811931891895429009}{1824468037770867911215483584823680077888} a^{15} - \frac{27934410550923516233659371673031638975}{1824468037770867911215483584823680077888} a^{13} - \frac{50568809182262617613846861577634296733}{1824468037770867911215483584823680077888} a^{11} - \frac{6347059904524269377783386574992440195}{114029252360679244450967724051480004868} a^{9} - \frac{7573857719159697499639664608865460361}{456117009442716977803870896205920019472} a^{7} - \frac{12831914381586293689293078810953931}{82570059638435368900049039863490228} a^{5} + \frac{12467242672483215209089957442205}{119580100852187355394712584885576} a^{3} + \frac{6898427548967347129303269671}{43294750489568195291351406548} a$

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:  \( 498475164989000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1013:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3686400
The 114 conjugacy class representatives for t20n1013 are not computed
Character table for t20n1013 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.109268775200000.1

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 32 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
5Data not computed
$13$13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.5.0.1$x^{5} - x + 11$$1$$5$$0$$C_5$$[\ ]^{5}$
29.5.0.1$x^{5} - x + 11$$1$$5$$0$$C_5$$[\ ]^{5}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.6.4.3$x^{6} + 713 x^{3} + 138384$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
31.8.0.1$x^{8} - x + 22$$1$$8$$0$$C_8$$[\ ]^{8}$
1381Data not computed