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Properties

Label 20.16.2653220688...9904.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{50}\cdot 31^{6}\cdot 227^{4}$
Root discriminant $46.90$
Ramified primes $2, 31, 227$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1966, -24224, 92420, -136008, 7804, 225120, -279636, 92400, 95949, -118164, 39272, 18888, -22121, 5072, 2778, -1688, 51, 152, -26, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 26*x^18 + 152*x^17 + 51*x^16 - 1688*x^15 + 2778*x^14 + 5072*x^13 - 22121*x^12 + 18888*x^11 + 39272*x^10 - 118164*x^9 + 95949*x^8 + 92400*x^7 - 279636*x^6 + 225120*x^5 + 7804*x^4 - 136008*x^3 + 92420*x^2 - 24224*x + 1966)
 
gp: K = bnfinit(x^20 - 4*x^19 - 26*x^18 + 152*x^17 + 51*x^16 - 1688*x^15 + 2778*x^14 + 5072*x^13 - 22121*x^12 + 18888*x^11 + 39272*x^10 - 118164*x^9 + 95949*x^8 + 92400*x^7 - 279636*x^6 + 225120*x^5 + 7804*x^4 - 136008*x^3 + 92420*x^2 - 24224*x + 1966, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 26 x^{18} + 152 x^{17} + 51 x^{16} - 1688 x^{15} + 2778 x^{14} + 5072 x^{13} - 22121 x^{12} + 18888 x^{11} + 39272 x^{10} - 118164 x^{9} + 95949 x^{8} + 92400 x^{7} - 279636 x^{6} + 225120 x^{5} + 7804 x^{4} - 136008 x^{3} + 92420 x^{2} - 24224 x + 1966 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2653220688038813933780612524539904=2^{50}\cdot 31^{6}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.90$
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:  $2, 31, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{31} a^{18} - \frac{3}{31} a^{17} - \frac{6}{31} a^{16} + \frac{15}{31} a^{15} - \frac{10}{31} a^{14} + \frac{11}{31} a^{13} - \frac{14}{31} a^{12} + \frac{10}{31} a^{11} + \frac{11}{31} a^{10} + \frac{2}{31} a^{9} + \frac{2}{31} a^{8} - \frac{6}{31} a^{7} + \frac{13}{31} a^{6} - \frac{12}{31} a^{5} - \frac{8}{31} a^{4} - \frac{7}{31} a^{3} - \frac{13}{31} a^{2} + \frac{1}{31} a - \frac{10}{31}$, $\frac{1}{4459814519578605181477397489} a^{19} - \frac{6086315099534296705952992}{4459814519578605181477397489} a^{18} + \frac{475646996307053005150974223}{4459814519578605181477397489} a^{17} - \frac{1128959120208330300993429166}{4459814519578605181477397489} a^{16} + \frac{2001429525561256110855767486}{4459814519578605181477397489} a^{15} + \frac{252902865102965786755591151}{4459814519578605181477397489} a^{14} + \frac{911352965633030097265847434}{4459814519578605181477397489} a^{13} + \frac{657039515643791904933123300}{4459814519578605181477397489} a^{12} - \frac{805348849403398514321258508}{4459814519578605181477397489} a^{11} - \frac{1184832505195306145703925860}{4459814519578605181477397489} a^{10} - \frac{487233693879032737970476966}{4459814519578605181477397489} a^{9} + \frac{1134940793073487378531949683}{4459814519578605181477397489} a^{8} - \frac{1075806964819087968832643554}{4459814519578605181477397489} a^{7} + \frac{1779299244492949253666697065}{4459814519578605181477397489} a^{6} - \frac{189092901336683564942837743}{4459814519578605181477397489} a^{5} - \frac{1932206724178077040220560835}{4459814519578605181477397489} a^{4} - \frac{1693104795670423690633259950}{4459814519578605181477397489} a^{3} + \frac{2181374656607320290762298299}{4459814519578605181477397489} a^{2} + \frac{809757978084652198848090746}{4459814519578605181477397489} a - \frac{309314547124882362228340598}{4459814519578605181477397489}$

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:  $17$
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:  \( 27603234914.0 \) (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

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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 $16{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }$ $16{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.8.24.7$x^{8} + 8 x^{7} + 12 x^{6} + 10 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 14$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.12.26.64$x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{6} - 2 x^{4} + 4 x^{3} + 2$$12$$1$$26$$S_3 \times C_2^2$$[2, 3]_{3}^{2}$
31Data not computed
227Data not computed