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Properties

Label 20.0.69504083479...3616.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 61^{4}\cdot 397^{4}\cdot 439^{2}$
Root discriminant $27.68$
Ramified primes $2, 61, 397, 439$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 20T368

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, 0, -512, 0, 704, 0, -80, 0, 124, 0, 29, 0, 31, 0, -5, 0, 11, 0, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^18 + 11*x^16 - 5*x^14 + 31*x^12 + 29*x^10 + 124*x^8 - 80*x^6 + 704*x^4 - 512*x^2 + 1024)
 
gp: K = bnfinit(x^20 - 2*x^18 + 11*x^16 - 5*x^14 + 31*x^12 + 29*x^10 + 124*x^8 - 80*x^6 + 704*x^4 - 512*x^2 + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{18} + 11 x^{16} - 5 x^{14} + 31 x^{12} + 29 x^{10} + 124 x^{8} - 80 x^{6} + 704 x^{4} - 512 x^{2} + 1024 \)

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:  \(69504083479956608292792303616=2^{20}\cdot 61^{4}\cdot 397^{4}\cdot 439^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.68$
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:  $2, 61, 397, 439$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{11} + \frac{3}{8} a^{9} + \frac{3}{8} a^{7} - \frac{1}{8} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{14} - \frac{1}{8} a^{12} - \frac{5}{16} a^{10} - \frac{5}{16} a^{8} - \frac{1}{16} a^{6} - \frac{3}{16} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{15} - \frac{1}{16} a^{13} - \frac{5}{32} a^{11} - \frac{5}{32} a^{9} + \frac{15}{32} a^{7} + \frac{13}{32} a^{5} + \frac{3}{8} a^{3}$, $\frac{1}{64} a^{16} - \frac{1}{32} a^{14} - \frac{5}{64} a^{12} + \frac{27}{64} a^{10} - \frac{17}{64} a^{8} - \frac{19}{64} a^{6} + \frac{3}{16} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{128} a^{17} - \frac{1}{64} a^{15} - \frac{5}{128} a^{13} + \frac{27}{128} a^{11} - \frac{17}{128} a^{9} - \frac{19}{128} a^{7} + \frac{3}{32} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{4352} a^{18} - \frac{1}{128} a^{16} + \frac{11}{4352} a^{14} - \frac{21}{256} a^{12} - \frac{513}{4352} a^{10} - \frac{2051}{4352} a^{8} - \frac{9}{64} a^{6} - \frac{73}{272} a^{4} + \frac{1}{4} a^{2} - \frac{2}{17}$, $\frac{1}{8704} a^{19} - \frac{1}{256} a^{17} + \frac{11}{8704} a^{15} - \frac{21}{512} a^{13} - \frac{513}{8704} a^{11} - \frac{2051}{8704} a^{9} + \frac{55}{128} a^{7} + \frac{199}{544} a^{5} + \frac{1}{8} a^{3} + \frac{15}{34} a$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( \frac{519}{8704} a^{19} - \frac{23}{256} a^{17} + \frac{3261}{8704} a^{15} + \frac{109}{512} a^{13} + \frac{4393}{8704} a^{11} + \frac{8027}{8704} a^{9} + \frac{733}{128} a^{7} - \frac{823}{136} a^{5} + \frac{123}{8} a^{3} - \frac{69}{34} a \) (order $4$)
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:  \( 2515406.93634 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T368:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 72 conjugacy class representatives for t20n368 are not computed
Character table for t20n368 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.5.24217.1, 10.10.263636271176704.1, 10.0.600538203136.1, 10.0.257457296071.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$61$61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
397Data not computed
439Data not computed