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Properties

Label 20.4.14012498575...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{38}\cdot 5^{14}\cdot 17^{4}$
Root discriminant $20.29$
Ramified primes $2, 5, 17$
Class number $1$
Class group Trivial
Galois group $D_5\wr C_2:C_2$ (as 20T96)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 12, 36, 24, 25, 32, 110, 16, 55, -128, 108, 28, -139, 140, -52, -4, 35, -24, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 14*x^18 - 24*x^17 + 35*x^16 - 4*x^15 - 52*x^14 + 140*x^13 - 139*x^12 + 28*x^11 + 108*x^10 - 128*x^9 + 55*x^8 + 16*x^7 + 110*x^6 + 32*x^5 + 25*x^4 + 24*x^3 + 36*x^2 + 12*x - 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 14*x^18 - 24*x^17 + 35*x^16 - 4*x^15 - 52*x^14 + 140*x^13 - 139*x^12 + 28*x^11 + 108*x^10 - 128*x^9 + 55*x^8 + 16*x^7 + 110*x^6 + 32*x^5 + 25*x^4 + 24*x^3 + 36*x^2 + 12*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 14 x^{18} - 24 x^{17} + 35 x^{16} - 4 x^{15} - 52 x^{14} + 140 x^{13} - 139 x^{12} + 28 x^{11} + 108 x^{10} - 128 x^{9} + 55 x^{8} + 16 x^{7} + 110 x^{6} + 32 x^{5} + 25 x^{4} + 24 x^{3} + 36 x^{2} + 12 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(140124985753600000000000000=2^{38}\cdot 5^{14}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.29$
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:  $2, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{3}$, $\frac{1}{40} a^{16} - \frac{1}{5} a^{15} + \frac{1}{8} a^{14} - \frac{1}{4} a^{13} + \frac{1}{40} a^{12} + \frac{1}{20} a^{10} + \frac{1}{5} a^{9} + \frac{3}{20} a^{8} - \frac{3}{10} a^{7} + \frac{2}{5} a^{6} - \frac{3}{10} a^{5} - \frac{11}{40} a^{4} + \frac{13}{40} a^{2} + \frac{3}{20} a - \frac{1}{40}$, $\frac{1}{40} a^{17} + \frac{1}{40} a^{15} - \frac{1}{4} a^{14} + \frac{1}{40} a^{13} + \frac{1}{5} a^{12} + \frac{1}{20} a^{11} + \frac{1}{10} a^{10} - \frac{1}{4} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{6} + \frac{13}{40} a^{5} + \frac{3}{10} a^{4} - \frac{7}{40} a^{3} + \frac{1}{4} a^{2} + \frac{7}{40} a + \frac{3}{10}$, $\frac{1}{1360} a^{18} + \frac{1}{272} a^{17} - \frac{7}{680} a^{16} - \frac{1}{272} a^{15} - \frac{9}{85} a^{14} - \frac{1}{80} a^{13} + \frac{107}{1360} a^{12} + \frac{67}{680} a^{11} + \frac{13}{68} a^{10} + \frac{203}{680} a^{9} - \frac{53}{136} a^{8} + \frac{19}{340} a^{7} - \frac{147}{1360} a^{6} + \frac{217}{1360} a^{5} - \frac{131}{680} a^{4} - \frac{77}{272} a^{3} + \frac{261}{680} a^{2} + \frac{1}{80} a + \frac{87}{272}$, $\frac{1}{12217977140560} a^{19} + \frac{193615269}{610898857028} a^{18} - \frac{152674158811}{12217977140560} a^{17} + \frac{146203724497}{12217977140560} a^{16} - \frac{824420627307}{12217977140560} a^{15} - \frac{2374555532977}{12217977140560} a^{14} - \frac{91727267289}{610898857028} a^{13} + \frac{142530934783}{2443595428112} a^{12} + \frac{1206335106583}{6108988570280} a^{11} - \frac{1396036472479}{6108988570280} a^{10} + \frac{101855039127}{1527247142570} a^{9} + \frac{2335595119503}{6108988570280} a^{8} + \frac{1940155772929}{12217977140560} a^{7} - \frac{1395546087217}{3054494285140} a^{6} - \frac{653726443847}{12217977140560} a^{5} + \frac{839833316269}{12217977140560} a^{4} - \frac{5537578193029}{12217977140560} a^{3} - \frac{4713585554697}{12217977140560} a^{2} + \frac{983279351489}{6108988570280} a - \frac{759681010451}{12217977140560}$

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:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 254286.481405 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\wr C_2:C_2$ (as 20T96):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 16 conjugacy class representatives for $D_5\wr C_2:C_2$
Character table for $D_5\wr C_2:C_2$

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 10.2.2367488000000.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 25 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$