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Properties

Label 20.8.31427919143...8329.2
Degree $20$
Signature $[8, 6]$
Discriminant $3^{6}\cdot 401^{11}$
Root discriminant $37.57$
Ramified primes $3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T350

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13347, 27807, -25253, -5316, 28375, -12849, 6893, -21274, 17213, 510, -10371, 10722, -6439, 1957, 251, -637, 383, -137, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 36*x^18 - 137*x^17 + 383*x^16 - 637*x^15 + 251*x^14 + 1957*x^13 - 6439*x^12 + 10722*x^11 - 10371*x^10 + 510*x^9 + 17213*x^8 - 21274*x^7 + 6893*x^6 - 12849*x^5 + 28375*x^4 - 5316*x^3 - 25253*x^2 + 27807*x - 13347)
 
gp: K = bnfinit(x^20 - 8*x^19 + 36*x^18 - 137*x^17 + 383*x^16 - 637*x^15 + 251*x^14 + 1957*x^13 - 6439*x^12 + 10722*x^11 - 10371*x^10 + 510*x^9 + 17213*x^8 - 21274*x^7 + 6893*x^6 - 12849*x^5 + 28375*x^4 - 5316*x^3 - 25253*x^2 + 27807*x - 13347, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 36 x^{18} - 137 x^{17} + 383 x^{16} - 637 x^{15} + 251 x^{14} + 1957 x^{13} - 6439 x^{12} + 10722 x^{11} - 10371 x^{10} + 510 x^{9} + 17213 x^{8} - 21274 x^{7} + 6893 x^{6} - 12849 x^{5} + 28375 x^{4} - 5316 x^{3} - 25253 x^{2} + 27807 x - 13347 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(31427919143590467585816658408329=3^{6}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.57$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{3} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{10} - \frac{1}{3} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{1}{6} a^{11} - \frac{1}{12} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{12} a^{7} - \frac{1}{4} a^{6} + \frac{1}{12} a^{5} + \frac{1}{3} a^{4} + \frac{5}{12} a^{3} + \frac{1}{12} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{12} a^{16} - \frac{1}{12} a^{14} - \frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{4} a^{10} - \frac{5}{12} a^{9} - \frac{5}{12} a^{8} + \frac{5}{12} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{5}{12} a^{4} + \frac{1}{4} a^{3} + \frac{5}{12} a^{2} - \frac{5}{12} a - \frac{1}{2}$, $\frac{1}{12} a^{17} - \frac{1}{12} a^{14} - \frac{1}{12} a^{13} + \frac{1}{12} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{6} a^{7} - \frac{1}{12} a^{6} + \frac{1}{6} a^{5} - \frac{5}{12} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{5}{12} a + \frac{1}{4}$, $\frac{1}{36} a^{18} - \frac{1}{36} a^{17} + \frac{1}{18} a^{14} - \frac{1}{18} a^{13} + \frac{1}{18} a^{12} - \frac{5}{36} a^{11} - \frac{7}{36} a^{10} + \frac{1}{12} a^{9} - \frac{1}{36} a^{8} - \frac{1}{9} a^{7} + \frac{5}{18} a^{5} - \frac{5}{12} a^{4} + \frac{13}{36} a^{3} - \frac{4}{9} a^{2} - \frac{1}{12} a - \frac{1}{2}$, $\frac{1}{25187591174043687636288483841346028} a^{19} + \frac{42555355229378051424188776557607}{4197931862340614606048080640224338} a^{18} - \frac{99761466422906035866193171500913}{6296897793510921909072120960336507} a^{17} + \frac{13863913221471212203540891490303}{8395863724681229212096161280448676} a^{16} - \frac{171464848355016578128733050722532}{6296897793510921909072120960336507} a^{15} + \frac{153890078231469021216639313795015}{2098965931170307303024040320112169} a^{14} - \frac{14400864479471396089004462156245}{1399310620780204868682693546741446} a^{13} + \frac{126353207822293008816308855970195}{2798621241560409737365387093482892} a^{12} - \frac{780039656992645983443327345160563}{4197931862340614606048080640224338} a^{11} - \frac{1210466739983933588343180386622127}{25187591174043687636288483841346028} a^{10} + \frac{3619775962999019739099802191657089}{25187591174043687636288483841346028} a^{9} - \frac{1285257952155340825397353905989957}{6296897793510921909072120960336507} a^{8} + \frac{11122317647915256641385018651986753}{25187591174043687636288483841346028} a^{7} - \frac{766285803057213238290397926618373}{12593795587021843818144241920673014} a^{6} + \frac{11874330123496773771703114307052631}{25187591174043687636288483841346028} a^{5} + \frac{5352182814495410767826070031593365}{12593795587021843818144241920673014} a^{4} - \frac{503580516814730324300780583151223}{2098965931170307303024040320112169} a^{3} - \frac{2789359912350427506484615368642661}{6296897793510921909072120960336507} a^{2} + \frac{3273019346472471896393855186669339}{8395863724681229212096161280448676} a + \frac{327845722091787587420546003475755}{2798621241560409737365387093482892}$

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:  $13$
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:  \( 218032726.126 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T350:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n350 are not computed
Character table for t20n350 is not computed

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
401Data not computed