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Properties

Label 20.0.75582720000...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 5^{23}$
Root discriminant $31.18$
Ramified primes $2, 3, 5$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2880, 0, -2400, 0, 2080, 0, 6120, 0, -2680, 0, -270, 0, 165, 0, 60, 0, -25, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 25*x^16 + 60*x^14 + 165*x^12 - 270*x^10 - 2680*x^8 + 6120*x^6 + 2080*x^4 - 2400*x^2 + 2880)
 
gp: K = bnfinit(x^20 - 25*x^16 + 60*x^14 + 165*x^12 - 270*x^10 - 2680*x^8 + 6120*x^6 + 2080*x^4 - 2400*x^2 + 2880, 1)
 

Normalized defining polynomial

\( x^{20} - 25 x^{16} + 60 x^{14} + 165 x^{12} - 270 x^{10} - 2680 x^{8} + 6120 x^{6} + 2080 x^{4} - 2400 x^{2} + 2880 \)

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:  \(755827200000000000000000000000=2^{30}\cdot 3^{10}\cdot 5^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.18$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{10} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2}$, $\frac{1}{12} a^{11} - \frac{1}{6} a^{7} - \frac{1}{4} a^{5} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{24} a^{12} + \frac{1}{24} a^{8} - \frac{1}{4} a^{6} + \frac{7}{24} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{24} a^{13} + \frac{1}{24} a^{9} - \frac{1}{4} a^{7} + \frac{7}{24} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{48} a^{14} - \frac{1}{48} a^{10} + \frac{5}{48} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{4} - \frac{5}{12} a^{2}$, $\frac{1}{48} a^{15} - \frac{1}{48} a^{11} + \frac{5}{48} a^{7} + \frac{3}{8} a^{5} - \frac{1}{2} a^{4} - \frac{5}{12} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{1947264} a^{16} + \frac{247}{324544} a^{14} - \frac{1}{48} a^{13} + \frac{14447}{1947264} a^{12} + \frac{40045}{973632} a^{10} + \frac{5}{48} a^{9} + \frac{192869}{1947264} a^{8} - \frac{5231}{44256} a^{6} + \frac{11}{48} a^{5} + \frac{60157}{486816} a^{4} + \frac{3}{8} a^{3} - \frac{2525}{60852} a^{2} + \frac{1}{4} a + \frac{10073}{40568}$, $\frac{1}{3894528} a^{17} + \frac{247}{649088} a^{15} - \frac{1}{96} a^{14} + \frac{14447}{3894528} a^{13} - \frac{13697}{649088} a^{11} - \frac{1}{32} a^{10} - \frac{293947}{3894528} a^{9} - \frac{1}{8} a^{8} - \frac{2973}{29504} a^{7} + \frac{5}{32} a^{6} - \frac{426659}{973632} a^{5} - \frac{7}{16} a^{4} + \frac{12681}{40568} a^{3} + \frac{1}{24} a^{2} + \frac{10073}{81136} a - \frac{1}{2}$, $\frac{1}{1756432128} a^{18} + \frac{85}{439108032} a^{16} - \frac{1}{96} a^{15} + \frac{17551235}{1756432128} a^{14} - \frac{1}{48} a^{13} - \frac{1766515}{109777008} a^{12} - \frac{1}{32} a^{11} + \frac{41874265}{1756432128} a^{10} - \frac{1}{48} a^{9} + \frac{12353215}{878216064} a^{8} + \frac{5}{32} a^{7} + \frac{26041645}{146369344} a^{6} - \frac{5}{24} a^{5} + \frac{73936529}{219554016} a^{4} - \frac{1}{12} a^{3} + \frac{4836457}{9979728} a^{2} - \frac{1}{4} a - \frac{5629979}{18296168}$, $\frac{1}{3512864256} a^{19} + \frac{85}{878216064} a^{17} - \frac{19041101}{3512864256} a^{15} - \frac{1766515}{219554016} a^{13} - \frac{1}{48} a^{12} - \frac{67902743}{3512864256} a^{11} + \frac{12353215}{1756432128} a^{9} + \frac{5}{48} a^{8} + \frac{35189729}{292738688} a^{7} + \frac{46492277}{439108032} a^{5} - \frac{13}{48} a^{4} + \frac{226079}{6653152} a^{3} - \frac{1}{8} a^{2} + \frac{12666189}{36592336} a + \frac{1}{4}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
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:  \( 4833940.00984 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.72000.2, 5.1.1800000.1 x5, 10.2.16200000000000.2 x5

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

Sibling fields

Degree 5 sibling: 5.1.1800000.1
Degree 10 sibling: 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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5Data not computed