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Properties

Label 20.0.67029370964...6352.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{33}\cdot 727^{8}$
Root discriminant $43.78$
Ramified primes $2, 727$
Class number $11$ (GRH)
Class group $[11]$ (GRH)
Galois group 20T525

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3200, 0, -10912, 0, 20004, 0, -8880, 0, -2335, 0, 2540, 0, -421, 0, -112, 0, 63, 0, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 12*x^18 + 63*x^16 - 112*x^14 - 421*x^12 + 2540*x^10 - 2335*x^8 - 8880*x^6 + 20004*x^4 - 10912*x^2 + 3200)
 
gp: K = bnfinit(x^20 - 12*x^18 + 63*x^16 - 112*x^14 - 421*x^12 + 2540*x^10 - 2335*x^8 - 8880*x^6 + 20004*x^4 - 10912*x^2 + 3200, 1)
 

Normalized defining polynomial

\( x^{20} - 12 x^{18} + 63 x^{16} - 112 x^{14} - 421 x^{12} + 2540 x^{10} - 2335 x^{8} - 8880 x^{6} + 20004 x^{4} - 10912 x^{2} + 3200 \)

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:  \(670293709640099417352021424996352=2^{33}\cdot 727^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.78$
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:  $2, 727$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} + \frac{3}{8} a^{5} - \frac{1}{2} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{6}$, $\frac{1}{80} a^{15} - \frac{1}{16} a^{14} + \frac{1}{40} a^{13} - \frac{1}{10} a^{11} + \frac{1}{10} a^{9} - \frac{1}{4} a^{8} - \frac{17}{80} a^{7} - \frac{3}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{10} a^{3} - \frac{1}{4} a^{2} + \frac{2}{5} a$, $\frac{1}{80} a^{16} - \frac{3}{80} a^{14} + \frac{1}{40} a^{12} + \frac{1}{10} a^{10} - \frac{17}{80} a^{8} - \frac{1}{16} a^{6} - \frac{1}{2} a^{5} - \frac{1}{40} a^{4} + \frac{2}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{160} a^{17} - \frac{1}{160} a^{15} - \frac{1}{40} a^{13} + \frac{3}{40} a^{11} - \frac{1}{160} a^{9} + \frac{1}{160} a^{7} + \frac{17}{40} a^{5} - \frac{3}{40} a^{3} + \frac{2}{5} a$, $\frac{1}{76560059004800} a^{18} + \frac{5582363607}{15312011800960} a^{16} - \frac{1083664780773}{19140014751200} a^{14} + \frac{709235342941}{19140014751200} a^{12} - \frac{1}{8} a^{11} + \frac{4605818895687}{76560059004800} a^{10} - \frac{1}{8} a^{9} + \frac{5323392635429}{76560059004800} a^{8} + \frac{1}{8} a^{7} - \frac{1802356323193}{19140014751200} a^{6} - \frac{1}{8} a^{5} - \frac{1039713771313}{2734287821600} a^{4} + \frac{1}{4} a^{3} + \frac{651291550003}{2392501843900} a^{2} + \frac{780705036}{23925018439}$, $\frac{1}{76560059004800} a^{19} + \frac{5582363607}{15312011800960} a^{17} + \frac{112586141177}{19140014751200} a^{15} - \frac{1}{16} a^{14} + \frac{709235342941}{19140014751200} a^{13} + \frac{4605818895687}{76560059004800} a^{11} - \frac{1}{8} a^{10} + \frac{5323392635429}{76560059004800} a^{9} + \frac{1}{8} a^{8} - \frac{2998607245143}{19140014751200} a^{7} - \frac{1}{16} a^{6} - \frac{1039713771313}{2734287821600} a^{5} + \frac{1}{8} a^{4} + \frac{651291550003}{2392501843900} a^{3} + \frac{780705036}{23925018439} a$

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

Class group and class number

$C_{11}$, which has order $11$ (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:  \( -\frac{252781}{2481687488} a^{18} + \frac{3705333}{2481687488} a^{16} - \frac{682615}{77552734} a^{14} + \frac{12322587}{620421872} a^{12} + \frac{123579525}{2481687488} a^{10} - \frac{1001121629}{2481687488} a^{8} + \frac{173213157}{310210936} a^{6} + \frac{135156029}{88631696} a^{4} - \frac{586744371}{155105468} a^{2} + \frac{48573319}{38776367} \) (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:  \( 2087758442.52 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.5.8456464.1, 10.0.286047133533184.1

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 $20$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ $20$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ $20$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.9.1$x^{4} + 6 x^{2} + 2$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
727Data not computed