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Properties

Label 20.0.64998372267...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{38}\cdot 3^{18}\cdot 5^{14}$
Root discriminant $30.95$
Ramified primes $2, 3, 5$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![576, 0, -288, 0, 288, 0, 720, 0, 612, 0, 174, 0, 12, 0, 60, 0, 48, 0, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 12*x^18 + 48*x^16 + 60*x^14 + 12*x^12 + 174*x^10 + 612*x^8 + 720*x^6 + 288*x^4 - 288*x^2 + 576)
 
gp: K = bnfinit(x^20 + 12*x^18 + 48*x^16 + 60*x^14 + 12*x^12 + 174*x^10 + 612*x^8 + 720*x^6 + 288*x^4 - 288*x^2 + 576, 1)
 

Normalized defining polynomial

\( x^{20} + 12 x^{18} + 48 x^{16} + 60 x^{14} + 12 x^{12} + 174 x^{10} + 612 x^{8} + 720 x^{6} + 288 x^{4} - 288 x^{2} + 576 \)

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:  \(649983722677862400000000000000=2^{38}\cdot 3^{18}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.95$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{6} a^{10}$, $\frac{1}{6} a^{11}$, $\frac{1}{12} a^{12} - \frac{1}{2} a^{2}$, $\frac{1}{24} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{48} a^{14} - \frac{1}{24} a^{12} + \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{8} a^{4} - \frac{1}{2}$, $\frac{1}{96} a^{15} - \frac{1}{48} a^{13} + \frac{1}{8} a^{9} - \frac{1}{8} a^{7} + \frac{1}{16} a^{5} - \frac{1}{4} a$, $\frac{1}{3648} a^{16} - \frac{3}{304} a^{14} - \frac{23}{912} a^{12} - \frac{5}{912} a^{10} + \frac{13}{304} a^{8} - \frac{59}{608} a^{6} - \frac{17}{304} a^{4} + \frac{23}{152} a^{2} + \frac{21}{76}$, $\frac{1}{3648} a^{17} + \frac{1}{1824} a^{15} - \frac{1}{228} a^{13} - \frac{5}{912} a^{11} + \frac{51}{304} a^{9} + \frac{169}{608} a^{7} - \frac{75}{152} a^{5} + \frac{61}{152} a^{3} - \frac{9}{19} a$, $\frac{1}{56281344} a^{18} + \frac{149}{1340032} a^{16} + \frac{32841}{4690112} a^{14} - \frac{67509}{4690112} a^{12} + \frac{174073}{14070336} a^{10} + \frac{217697}{9380224} a^{8} - \frac{685493}{2345056} a^{6} - \frac{245663}{586264} a^{4} - \frac{1663}{83752} a^{2} + \frac{280179}{586264}$, $\frac{1}{168844032} a^{19} + \frac{149}{4020096} a^{17} + \frac{10947}{4690112} a^{15} - \frac{22503}{4690112} a^{13} - \frac{723661}{14070336} a^{11} - \frac{9162527}{28140672} a^{9} - \frac{1010183}{2345056} a^{7} - \frac{277309}{586264} a^{5} + \frac{27363}{83752} a^{3} + \frac{93393}{586264} a$

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

Class group and class number

$C_{4}$, 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:  \( 9660809.80023597 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-6}, \sqrt{10})\), 5.1.162000.1, 10.2.268738560000000.13, 10.0.393660000000.1, 10.0.161243136000000.6

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.19.17$x^{10} - 2 x^{4} + 4 x^{2} - 10$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
2.10.19.17$x^{10} - 2 x^{4} + 4 x^{2} - 10$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$3$3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$