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Properties

Label 20.12.5540004668...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{10}\cdot 13^{4}\cdot 29^{5}\cdot 31^{4}$
Root discriminant $34.45$
Ramified primes $2, 5, 13, 29, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1036

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -68, 296, -112, -1480, 1564, 2620, -3020, -4006, 4722, 3318, -4750, -738, 2128, -136, -454, 89, 43, -16, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 16*x^18 + 43*x^17 + 89*x^16 - 454*x^15 - 136*x^14 + 2128*x^13 - 738*x^12 - 4750*x^11 + 3318*x^10 + 4722*x^9 - 4006*x^8 - 3020*x^7 + 2620*x^6 + 1564*x^5 - 1480*x^4 - 112*x^3 + 296*x^2 - 68*x + 4)
 
gp: K = bnfinit(x^20 - x^19 - 16*x^18 + 43*x^17 + 89*x^16 - 454*x^15 - 136*x^14 + 2128*x^13 - 738*x^12 - 4750*x^11 + 3318*x^10 + 4722*x^9 - 4006*x^8 - 3020*x^7 + 2620*x^6 + 1564*x^5 - 1480*x^4 - 112*x^3 + 296*x^2 - 68*x + 4, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 16 x^{18} + 43 x^{17} + 89 x^{16} - 454 x^{15} - 136 x^{14} + 2128 x^{13} - 738 x^{12} - 4750 x^{11} + 3318 x^{10} + 4722 x^{9} - 4006 x^{8} - 3020 x^{7} + 2620 x^{6} + 1564 x^{5} - 1480 x^{4} - 112 x^{3} + 296 x^{2} - 68 x + 4 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5540004668440574658560000000000=2^{20}\cdot 5^{10}\cdot 13^{4}\cdot 29^{5}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.45$
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:  $2, 5, 13, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{9}$, $\frac{1}{23341387964687848809861670} a^{19} + \frac{2231275902094954371094263}{23341387964687848809861670} a^{18} - \frac{767528843030292729329627}{11670693982343924404930835} a^{17} + \frac{1226729782552746197374497}{23341387964687848809861670} a^{16} - \frac{936382423669298646027534}{11670693982343924404930835} a^{15} - \frac{2846736668556140801017298}{11670693982343924404930835} a^{14} - \frac{453977970452479871348007}{2334138796468784880986167} a^{13} + \frac{3101175541529198409172543}{23341387964687848809861670} a^{12} - \frac{1096687321391468490187681}{23341387964687848809861670} a^{11} - \frac{3089725176154408353254457}{11670693982343924404930835} a^{10} + \frac{5025891092867708859307376}{11670693982343924404930835} a^{9} + \frac{1214709463954903159527741}{4668277592937569761972334} a^{8} + \frac{141412566987514774777132}{11670693982343924404930835} a^{7} - \frac{3343255367419305825084032}{11670693982343924404930835} a^{6} - \frac{2682023904538312677479448}{11670693982343924404930835} a^{5} + \frac{270397282683876121414029}{2334138796468784880986167} a^{4} + \frac{255415466943805657118420}{2334138796468784880986167} a^{3} + \frac{1500355571391770906009224}{11670693982343924404930835} a^{2} - \frac{1581794860000344408979086}{11670693982343924404930835} a - \frac{935000251657665924097483}{11670693982343924404930835}$

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:  $15$
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:  \( 440483040.628 \) (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

\(\Q(\sqrt{5}) \), 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 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 $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
5Data not computed
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
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}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.6.5.1$x^{6} - 29$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
29.10.0.1$x^{10} + x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
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}$