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Properties

Label 20.12.2700994879...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{58}\cdot 5^{20}\cdot 7^{6}\cdot 17^{4}$
Root discriminant $117.92$
Ramified primes $2, 5, 7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T872

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![117649, 0, 168070, 0, -228095, 0, -205800, 0, 159250, 0, 13048, 0, -17330, 0, 1800, 0, 175, 0, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 30*x^18 + 175*x^16 + 1800*x^14 - 17330*x^12 + 13048*x^10 + 159250*x^8 - 205800*x^6 - 228095*x^4 + 168070*x^2 + 117649)
 
gp: K = bnfinit(x^20 - 30*x^18 + 175*x^16 + 1800*x^14 - 17330*x^12 + 13048*x^10 + 159250*x^8 - 205800*x^6 - 228095*x^4 + 168070*x^2 + 117649, 1)
 

Normalized defining polynomial

\( x^{20} - 30 x^{18} + 175 x^{16} + 1800 x^{14} - 17330 x^{12} + 13048 x^{10} + 159250 x^{8} - 205800 x^{6} - 228095 x^{4} + 168070 x^{2} + 117649 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(270099487931191892377600000000000000000000=2^{58}\cdot 5^{20}\cdot 7^{6}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $117.92$
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:  $2, 5, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{1}{7} a^{4} + \frac{3}{7} a^{2}$, $\frac{1}{7} a^{7} - \frac{1}{7} a^{5} + \frac{3}{7} a^{3}$, $\frac{1}{7} a^{8} + \frac{2}{7} a^{4} + \frac{3}{7} a^{2}$, $\frac{1}{7} a^{9} + \frac{2}{7} a^{5} + \frac{3}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{2}{7} a^{4} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{11} - \frac{2}{7} a^{5} + \frac{1}{7} a^{3}$, $\frac{1}{49} a^{12} - \frac{2}{49} a^{10} + \frac{1}{49} a^{6} - \frac{12}{49} a^{4}$, $\frac{1}{49} a^{13} - \frac{2}{49} a^{11} + \frac{1}{49} a^{7} - \frac{12}{49} a^{5}$, $\frac{1}{343} a^{14} - \frac{2}{343} a^{12} + \frac{3}{49} a^{10} - \frac{13}{343} a^{8} - \frac{5}{343} a^{6} - \frac{18}{49} a^{4} + \frac{3}{7} a^{2}$, $\frac{1}{343} a^{15} - \frac{2}{343} a^{13} + \frac{3}{49} a^{11} - \frac{13}{343} a^{9} - \frac{5}{343} a^{7} - \frac{18}{49} a^{5} + \frac{3}{7} a^{3}$, $\frac{1}{2401} a^{16} - \frac{2}{2401} a^{14} + \frac{3}{343} a^{12} - \frac{160}{2401} a^{10} + \frac{142}{2401} a^{8} + \frac{17}{343} a^{6} - \frac{11}{49} a^{4} + \frac{2}{7} a^{2}$, $\frac{1}{2401} a^{17} - \frac{2}{2401} a^{15} + \frac{3}{343} a^{13} - \frac{160}{2401} a^{11} + \frac{142}{2401} a^{9} + \frac{17}{343} a^{7} - \frac{11}{49} a^{5} + \frac{2}{7} a^{3}$, $\frac{1}{97917994806693551} a^{18} + \frac{8819416763701}{97917994806693551} a^{16} + \frac{1866448781194}{13988284972384793} a^{14} + \frac{723197691555725}{97917994806693551} a^{12} - \frac{4128905717143023}{97917994806693551} a^{10} + \frac{466997853113829}{13988284972384793} a^{8} + \frac{104996459252686}{1998326424626399} a^{6} + \frac{4632725488121}{285475203518057} a^{4} + \frac{518544502099}{40782171931151} a^{2} - \frac{1740984611604}{5826024561593}$, $\frac{1}{97917994806693551} a^{19} + \frac{8819416763701}{97917994806693551} a^{17} + \frac{1866448781194}{13988284972384793} a^{15} + \frac{723197691555725}{97917994806693551} a^{13} - \frac{4128905717143023}{97917994806693551} a^{11} + \frac{466997853113829}{13988284972384793} a^{9} + \frac{104996459252686}{1998326424626399} a^{7} + \frac{4632725488121}{285475203518057} a^{5} + \frac{518544502099}{40782171931151} a^{3} - \frac{1740984611604}{5826024561593} a$

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:  $15$
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:  \( 93408305777300 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T872:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 204800
The 116 conjugacy class representatives for t20n872 are not computed
Character table for t20n872 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.1479680000000000.1

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 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 R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.10.14$x^{10} + 10 x^{6} + 10 x^{5} + 100 x^{2} + 50 x + 25$$5$$2$$10$$(C_5^2 : C_4) : C_2$$[5/4, 5/4]_{4}^{2}$
5.10.10.14$x^{10} + 10 x^{6} + 10 x^{5} + 100 x^{2} + 50 x + 25$$5$$2$$10$$(C_5^2 : C_4) : C_2$$[5/4, 5/4]_{4}^{2}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$