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Properties

Label 20.4.89099712768...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 5^{20}\cdot 7^{6}\cdot 59^{4}$
Root discriminant $35.28$
Ramified primes $2, 5, 7, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![784, 0, 980, 0, -2415, 0, -5215, 0, -4080, 0, -2094, 0, -490, 0, -60, 0, 25, 0, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 5*x^18 + 25*x^16 - 60*x^14 - 490*x^12 - 2094*x^10 - 4080*x^8 - 5215*x^6 - 2415*x^4 + 980*x^2 + 784)
 
gp: K = bnfinit(x^20 + 5*x^18 + 25*x^16 - 60*x^14 - 490*x^12 - 2094*x^10 - 4080*x^8 - 5215*x^6 - 2415*x^4 + 980*x^2 + 784, 1)
 

Normalized defining polynomial

\( x^{20} + 5 x^{18} + 25 x^{16} - 60 x^{14} - 490 x^{12} - 2094 x^{10} - 4080 x^{8} - 5215 x^{6} - 2415 x^{4} + 980 x^{2} + 784 \)

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:  \(8909971276806250000000000000000=2^{16}\cdot 5^{20}\cdot 7^{6}\cdot 59^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.28$
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:  $2, 5, 7, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} - \frac{2}{7} a^{14} - \frac{3}{7} a^{12} + \frac{3}{7} a^{10} - \frac{1}{7} a^{6} + \frac{1}{7} a^{4}$, $\frac{1}{14} a^{17} + \frac{5}{14} a^{15} - \frac{3}{14} a^{13} - \frac{2}{7} a^{11} + \frac{3}{7} a^{7} - \frac{3}{7} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{13599500016055912} a^{18} - \frac{1}{28} a^{17} + \frac{816742086642637}{13599500016055912} a^{16} - \frac{5}{28} a^{15} - \frac{2799104716646191}{13599500016055912} a^{14} + \frac{3}{28} a^{13} - \frac{571356629604143}{3399875004013978} a^{12} + \frac{1}{7} a^{11} - \frac{3285852861184501}{6799750008027956} a^{10} - \frac{1}{2} a^{9} - \frac{3210623735327719}{6799750008027956} a^{8} - \frac{3}{14} a^{7} + \frac{322492983758111}{1699937502006989} a^{6} - \frac{2}{7} a^{5} - \frac{4231493662128303}{13599500016055912} a^{4} + \frac{1}{4} a^{3} - \frac{322416420000889}{1942785716579416} a^{2} + \frac{1}{4} a - \frac{33494507652331}{485696429144854}$, $\frac{1}{27199000032111824} a^{19} - \frac{22092967378153}{3885571433158832} a^{17} - \frac{1}{14} a^{16} - \frac{1093724144013533}{3885571433158832} a^{15} - \frac{5}{14} a^{14} - \frac{810671747475210}{1699937502006989} a^{13} + \frac{3}{14} a^{12} - \frac{1343067144605085}{13599500016055912} a^{11} + \frac{2}{7} a^{10} + \frac{3589126272700237}{13599500016055912} a^{9} + \frac{1293885842047819}{3399875004013978} a^{7} - \frac{3}{7} a^{6} - \frac{12002636528445967}{27199000032111824} a^{5} + \frac{3}{7} a^{4} - \frac{1293809278290597}{3885571433158832} a^{3} - \frac{1}{2} a^{2} + \frac{52338426730024}{242848214572427} a - \frac{1}{2}$

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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 34266856.1019 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.2.1665712890625.1

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 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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.8.0.1}{8} }$ R R ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ R

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
5Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{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}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
59Data not computed