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Properties

Label 20.0.64009258510...0112.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{44}\cdot 439\cdot 1663\cdot 2657^{4}$
Root discriminant $43.68$
Ramified primes $2, 439, 1663, 2657$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1036

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16924, -32600, 49668, -56792, 67754, -54220, 57740, -32248, 35017, -13008, 15572, -3560, 5113, -648, 1208, -72, 197, -4, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 20*x^18 - 4*x^17 + 197*x^16 - 72*x^15 + 1208*x^14 - 648*x^13 + 5113*x^12 - 3560*x^11 + 15572*x^10 - 13008*x^9 + 35017*x^8 - 32248*x^7 + 57740*x^6 - 54220*x^5 + 67754*x^4 - 56792*x^3 + 49668*x^2 - 32600*x + 16924)
 
gp: K = bnfinit(x^20 + 20*x^18 - 4*x^17 + 197*x^16 - 72*x^15 + 1208*x^14 - 648*x^13 + 5113*x^12 - 3560*x^11 + 15572*x^10 - 13008*x^9 + 35017*x^8 - 32248*x^7 + 57740*x^6 - 54220*x^5 + 67754*x^4 - 56792*x^3 + 49668*x^2 - 32600*x + 16924, 1)
 

Normalized defining polynomial

\( x^{20} + 20 x^{18} - 4 x^{17} + 197 x^{16} - 72 x^{15} + 1208 x^{14} - 648 x^{13} + 5113 x^{12} - 3560 x^{11} + 15572 x^{10} - 13008 x^{9} + 35017 x^{8} - 32248 x^{7} + 57740 x^{6} - 54220 x^{5} + 67754 x^{4} - 56792 x^{3} + 49668 x^{2} - 32600 x + 16924 \)

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:  \(640092585108940580184055833690112=2^{44}\cdot 439\cdot 1663\cdot 2657^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.68$
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:  $2, 439, 1663, 2657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{4379279850657456319225827047558} a^{19} + \frac{94102882619352758258420471338}{2189639925328728159612913523779} a^{18} - \frac{246415174471991773110284071434}{2189639925328728159612913523779} a^{17} - \frac{74582387513800781745540370222}{2189639925328728159612913523779} a^{16} + \frac{25067620093529838507278870047}{230488413192497701011885634082} a^{15} + \frac{512817832327332959399659952617}{2189639925328728159612913523779} a^{14} + \frac{56518534070670017831784410391}{115244206596248850505942817041} a^{13} + \frac{992733709951985306317251854313}{2189639925328728159612913523779} a^{12} + \frac{1748347743534525446468123649471}{4379279850657456319225827047558} a^{11} - \frac{263497974286888374068229860016}{2189639925328728159612913523779} a^{10} + \frac{773470243868506516678135848483}{2189639925328728159612913523779} a^{9} + \frac{228405273224152117674581863254}{2189639925328728159612913523779} a^{8} - \frac{853443807474886479915057623371}{4379279850657456319225827047558} a^{7} - \frac{700152814501902477966856425663}{2189639925328728159612913523779} a^{6} - \frac{454083843537419059855379251133}{2189639925328728159612913523779} a^{5} - \frac{274763058157248951621201034847}{2189639925328728159612913523779} a^{4} - \frac{1002360761042943121999013534002}{2189639925328728159612913523779} a^{3} - \frac{930166392966633935318918320590}{2189639925328728159612913523779} a^{2} + \frac{877189950511052372194953968667}{2189639925328728159612913523779} a - \frac{209353276208132098941999641583}{2189639925328728159612913523779}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 121129532.773 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1036:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 396 conjugacy class representatives for t20n1036 are not computed
Character table for t20n1036 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.6.925322313728.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.16.13$x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
2.12.28.65$x^{12} + 2 x^{10} + 4 x^{9} - 2 x^{8} + 4 x^{6} + 4 x^{5} + 2$$12$$1$$28$12T48$[2, 8/3, 8/3, 3]_{3}^{2}$
439Data not computed
1663Data not computed
2657Data not computed