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Properties

Label 20.14.2180059243...0000.1
Degree $20$
Signature $[14, 3]$
Discriminant $-\,2^{40}\cdot 5^{11}\cdot 67^{8}$
Root discriminant $52.11$
Ramified primes $2, 5, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T803

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-461, -18814, 5801, 132356, -121595, -170940, 174218, 125000, -103678, -80412, 41076, 34332, -11818, -7300, 1554, 688, 31, -22, -21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 21*x^18 - 22*x^17 + 31*x^16 + 688*x^15 + 1554*x^14 - 7300*x^13 - 11818*x^12 + 34332*x^11 + 41076*x^10 - 80412*x^9 - 103678*x^8 + 125000*x^7 + 174218*x^6 - 170940*x^5 - 121595*x^4 + 132356*x^3 + 5801*x^2 - 18814*x - 461)
 
gp: K = bnfinit(x^20 - 21*x^18 - 22*x^17 + 31*x^16 + 688*x^15 + 1554*x^14 - 7300*x^13 - 11818*x^12 + 34332*x^11 + 41076*x^10 - 80412*x^9 - 103678*x^8 + 125000*x^7 + 174218*x^6 - 170940*x^5 - 121595*x^4 + 132356*x^3 + 5801*x^2 - 18814*x - 461, 1)
 

Normalized defining polynomial

\( x^{20} - 21 x^{18} - 22 x^{17} + 31 x^{16} + 688 x^{15} + 1554 x^{14} - 7300 x^{13} - 11818 x^{12} + 34332 x^{11} + 41076 x^{10} - 80412 x^{9} - 103678 x^{8} + 125000 x^{7} + 174218 x^{6} - 170940 x^{5} - 121595 x^{4} + 132356 x^{3} + 5801 x^{2} - 18814 x - 461 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-21800592438355578532659200000000000=-\,2^{40}\cdot 5^{11}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.11$
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:  $2, 5, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{12} a^{16} - \frac{1}{12} a^{14} - \frac{1}{6} a^{13} - \frac{1}{6} a^{12} + \frac{1}{3} a^{11} - \frac{1}{12} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{5}{12} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} + \frac{1}{4} a^{2} - \frac{1}{6} a + \frac{1}{12}$, $\frac{1}{12} a^{17} - \frac{1}{12} a^{15} - \frac{1}{6} a^{14} - \frac{1}{6} a^{13} - \frac{1}{6} a^{12} - \frac{1}{12} a^{11} - \frac{1}{3} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{5}{12} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{4} a^{3} + \frac{1}{3} a^{2} + \frac{1}{12} a - \frac{1}{2}$, $\frac{1}{60} a^{18} + \frac{1}{60} a^{17} + \frac{1}{60} a^{16} - \frac{1}{4} a^{15} + \frac{1}{5} a^{14} + \frac{1}{6} a^{13} + \frac{11}{60} a^{12} + \frac{7}{20} a^{11} - \frac{13}{30} a^{10} - \frac{3}{10} a^{9} - \frac{23}{60} a^{8} + \frac{17}{60} a^{7} - \frac{2}{5} a^{6} - \frac{1}{30} a^{5} - \frac{13}{60} a^{4} - \frac{1}{4} a^{3} + \frac{17}{60} a^{2} - \frac{9}{20} a - \frac{1}{15}$, $\frac{1}{59386950241092687108614834912761189573260} a^{19} - \frac{124827839530301537168323516886329461087}{19795650080364229036204944970920396524420} a^{18} + \frac{96822971132526843705351985271604053487}{4948912520091057259051236242730099131105} a^{17} + \frac{419041955564903164249540440460771615241}{19795650080364229036204944970920396524420} a^{16} + \frac{15913697294102020558643950947140692799}{1799604552760384457836813179174581502220} a^{15} + \frac{603673706614724413024854843518626233808}{4948912520091057259051236242730099131105} a^{14} - \frac{167175640932529368588341475104007835669}{59386950241092687108614834912761189573260} a^{13} - \frac{6108527824010419088909451261253367836261}{59386950241092687108614834912761189573260} a^{12} - \frac{23737593332171634943041729969938988130003}{59386950241092687108614834912761189573260} a^{11} + \frac{3550186854522926985594459824231717569651}{14846737560273171777153708728190297393315} a^{10} - \frac{3525686407321273215749438464115983430099}{19795650080364229036204944970920396524420} a^{9} + \frac{11507766339139592120764502867844411419363}{59386950241092687108614834912761189573260} a^{8} - \frac{20223233574225369488821691135591283496153}{59386950241092687108614834912761189573260} a^{7} - \frac{4453849386027258890645254886173991269936}{14846737560273171777153708728190297393315} a^{6} + \frac{2966900983890276641594145314325493307677}{19795650080364229036204944970920396524420} a^{5} - \frac{3513580175861550086598710892204944982963}{19795650080364229036204944970920396524420} a^{4} - \frac{2223264864354223462170368845774577938604}{4948912520091057259051236242730099131105} a^{3} + \frac{4410916079028188339682836499075584315869}{59386950241092687108614834912761189573260} a^{2} - \frac{60562576013869395957403203330291364521}{359920910552076891567362635834916300444} a - \frac{6895827518875449880105293830573230324213}{14846737560273171777153708728190297393315}$

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:  $16$
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:  \( 45704084958.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T803:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 126 conjugacy class representatives for t20n803 are not computed
Character table for t20n803 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.5745920.1, 10.10.4126949580800000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$67$67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.6.4.1$x^{6} + 2345 x^{3} + 7756992$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
67.6.4.1$x^{6} + 2345 x^{3} + 7756992$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$