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Properties

Label 20.0.24677119028...6656.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{12}\cdot 3^{15}\cdot 13^{4}\cdot 43^{5}$
Root discriminant $14.78$
Ramified primes $2, 3, 13, 43$
Class number $1$
Class group Trivial
Galois group 20T658

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 36, -119, 300, -599, 993, -1366, 1603, -1656, 1533, -1242, 863, -514, 273, -137, 68, -31, 12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 12*x^18 - 31*x^17 + 68*x^16 - 137*x^15 + 273*x^14 - 514*x^13 + 863*x^12 - 1242*x^11 + 1533*x^10 - 1656*x^9 + 1603*x^8 - 1366*x^7 + 993*x^6 - 599*x^5 + 300*x^4 - 119*x^3 + 36*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 12*x^18 - 31*x^17 + 68*x^16 - 137*x^15 + 273*x^14 - 514*x^13 + 863*x^12 - 1242*x^11 + 1533*x^10 - 1656*x^9 + 1603*x^8 - 1366*x^7 + 993*x^6 - 599*x^5 + 300*x^4 - 119*x^3 + 36*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 12 x^{18} - 31 x^{17} + 68 x^{16} - 137 x^{15} + 273 x^{14} - 514 x^{13} + 863 x^{12} - 1242 x^{11} + 1533 x^{10} - 1656 x^{9} + 1603 x^{8} - 1366 x^{7} + 993 x^{6} - 599 x^{5} + 300 x^{4} - 119 x^{3} + 36 x^{2} - 8 x + 1 \)

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:  \(246771190286366549446656=2^{12}\cdot 3^{15}\cdot 13^{4}\cdot 43^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.78$
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:  $2, 3, 13, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{18113370815858} a^{19} - \frac{748466997961}{18113370815858} a^{18} - \frac{152548066722}{9056685407929} a^{17} - \frac{104356459395}{18113370815858} a^{16} + \frac{1035800590239}{9056685407929} a^{15} - \frac{31049821520}{9056685407929} a^{14} + \frac{8027470672459}{18113370815858} a^{13} - \frac{6474887172141}{18113370815858} a^{12} - \frac{3893730547201}{9056685407929} a^{11} + \frac{6686760329287}{18113370815858} a^{10} + \frac{8938347270481}{18113370815858} a^{9} - \frac{2673625519220}{9056685407929} a^{8} - \frac{5542132077587}{18113370815858} a^{7} + \frac{6169905401451}{18113370815858} a^{6} + \frac{2446563641277}{9056685407929} a^{5} + \frac{503078186314}{9056685407929} a^{4} - \frac{5207732739607}{18113370815858} a^{3} - \frac{40635131367}{87928984543} a^{2} + \frac{3954388903567}{18113370815858} a + \frac{1538255804947}{18113370815858}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{67312733}{22700138} a^{19} + \frac{122429821}{11350069} a^{18} - \frac{712987607}{22700138} a^{17} + \frac{1806291821}{22700138} a^{16} - \frac{1928396958}{11350069} a^{15} + \frac{3828400750}{11350069} a^{14} - \frac{7621214415}{11350069} a^{13} + \frac{28355035953}{22700138} a^{12} - \frac{46386376821}{22700138} a^{11} + \frac{64130004557}{22700138} a^{10} - \frac{37773776004}{11350069} a^{9} + \frac{38961554160}{11350069} a^{8} - \frac{36148147374}{11350069} a^{7} + \frac{58091361881}{22700138} a^{6} - \frac{38680209361}{22700138} a^{5} + \frac{10272327556}{11350069} a^{4} - \frac{8864173755}{22700138} a^{3} + \frac{1379732054}{11350069} a^{2} - \frac{305862350}{11350069} a + \frac{48252193}{11350069} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 11111.2700135 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T658:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 76 conjugacy class representatives for t20n658 are not computed
Character table for t20n658 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.1161.1, 10.0.4859704512.1

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 24 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 R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ R $20$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
3Data not computed
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.6.4.2$x^{6} - 13 x^{3} + 338$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
43Data not computed