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Properties

Label 20.20.7909954613...8125.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{10}\cdot 5^{17}\cdot 23^{4}\cdot 89^{4}$
Root discriminant $31.25$
Ramified primes $3, 5, 23, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\times S_5$ (as 20T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -28, 120, 343, -1872, -1585, 8947, 2142, -18435, 2813, 17547, -7662, -6723, 4504, 927, -1110, 30, 123, -17, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 17*x^18 + 123*x^17 + 30*x^16 - 1110*x^15 + 927*x^14 + 4504*x^13 - 6723*x^12 - 7662*x^11 + 17547*x^10 + 2813*x^9 - 18435*x^8 + 2142*x^7 + 8947*x^6 - 1585*x^5 - 1872*x^4 + 343*x^3 + 120*x^2 - 28*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 - 17*x^18 + 123*x^17 + 30*x^16 - 1110*x^15 + 927*x^14 + 4504*x^13 - 6723*x^12 - 7662*x^11 + 17547*x^10 + 2813*x^9 - 18435*x^8 + 2142*x^7 + 8947*x^6 - 1585*x^5 - 1872*x^4 + 343*x^3 + 120*x^2 - 28*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 17 x^{18} + 123 x^{17} + 30 x^{16} - 1110 x^{15} + 927 x^{14} + 4504 x^{13} - 6723 x^{12} - 7662 x^{11} + 17547 x^{10} + 2813 x^{9} - 18435 x^{8} + 2142 x^{7} + 8947 x^{6} - 1585 x^{5} - 1872 x^{4} + 343 x^{3} + 120 x^{2} - 28 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(790995461333388800811767578125=3^{10}\cdot 5^{17}\cdot 23^{4}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.25$
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:  $3, 5, 23, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{366723360113161} a^{19} + \frac{38662708795571}{366723360113161} a^{18} - \frac{103381480817965}{366723360113161} a^{17} + \frac{78622201941980}{366723360113161} a^{16} + \frac{56592142033739}{366723360113161} a^{15} + \frac{67708482144707}{366723360113161} a^{14} - \frac{93469255764209}{366723360113161} a^{13} + \frac{156021892882639}{366723360113161} a^{12} + \frac{7527755233583}{366723360113161} a^{11} + \frac{18049494477853}{366723360113161} a^{10} - \frac{145355843231020}{366723360113161} a^{9} - \frac{137665989231919}{366723360113161} a^{8} + \frac{93499882566903}{366723360113161} a^{7} + \frac{81550679477475}{366723360113161} a^{6} - \frac{81117744653400}{366723360113161} a^{5} + \frac{44383513678625}{366723360113161} a^{4} - \frac{41976901450825}{366723360113161} a^{3} + \frac{105581996614153}{366723360113161} a^{2} + \frac{47111130019654}{366723360113161} a + \frac{116721941908410}{366723360113161}$

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:  $19$
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:  \( 286844082.384 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times S_5$ (as 20T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 480
The 28 conjugacy class representatives for $C_4\times S_5$
Character table for $C_4\times S_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 5.5.767625.1, 10.10.2946240703125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: 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 $20$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.12.6.1$x^{12} - 243 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
5Data not computed
$23$23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
89Data not computed