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Properties

Label 20.10.9260662602...7616.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{26}\cdot 53^{14}$
Root discriminant $39.66$
Ramified primes $2, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T513

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-64, 0, -576, 0, 64, 0, 1568, 0, 28, 0, -1324, 0, -2, 0, 354, 0, -21, 0, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 12*x^18 - 21*x^16 + 354*x^14 - 2*x^12 - 1324*x^10 + 28*x^8 + 1568*x^6 + 64*x^4 - 576*x^2 - 64)
 
gp: K = bnfinit(x^20 - 12*x^18 - 21*x^16 + 354*x^14 - 2*x^12 - 1324*x^10 + 28*x^8 + 1568*x^6 + 64*x^4 - 576*x^2 - 64, 1)
 

Normalized defining polynomial

\( x^{20} - 12 x^{18} - 21 x^{16} + 354 x^{14} - 2 x^{12} - 1324 x^{10} + 28 x^{8} + 1568 x^{6} + 64 x^{4} - 576 x^{2} - 64 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-92606626027701221381101758447616=-\,2^{26}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.66$
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:  $2, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{14} - \frac{1}{8} a^{11} + \frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{16} a^{15} - \frac{1}{8} a^{12} + \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{16} + \frac{7}{64} a^{12} - \frac{1}{8} a^{11} - \frac{1}{32} a^{10} - \frac{1}{8} a^{9} - \frac{7}{32} a^{8} - \frac{3}{16} a^{6} - \frac{1}{4} a^{5} + \frac{5}{16} a^{4} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{128} a^{17} - \frac{1}{128} a^{13} - \frac{1}{8} a^{12} - \frac{5}{64} a^{11} + \frac{9}{64} a^{9} + \frac{1}{8} a^{8} - \frac{7}{32} a^{7} - \frac{1}{4} a^{6} - \frac{3}{32} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{5576112896} a^{18} - \frac{7174453}{2788056448} a^{16} - \frac{1}{32} a^{15} + \frac{106919791}{5576112896} a^{14} - \frac{1}{16} a^{13} + \frac{8938241}{174253528} a^{12} - \frac{3}{32} a^{11} - \frac{227827629}{2788056448} a^{10} - \frac{1}{8} a^{9} - \frac{77917065}{348507056} a^{8} - \frac{1}{16} a^{7} + \frac{94380811}{1394028224} a^{6} - \frac{1}{4} a^{5} + \frac{2710363}{697014112} a^{4} + \frac{3}{8} a^{3} + \frac{161819229}{348507056} a^{2} - \frac{1}{4} a - \frac{48755493}{174253528}$, $\frac{1}{11152225792} a^{19} - \frac{7174453}{5576112896} a^{17} - \frac{1}{128} a^{16} + \frac{106919791}{11152225792} a^{15} - \frac{1}{32} a^{14} - \frac{6421725}{174253528} a^{13} - \frac{7}{128} a^{12} - \frac{227827629}{5576112896} a^{11} - \frac{1}{64} a^{10} - \frac{34353683}{697014112} a^{9} + \frac{15}{64} a^{8} + \frac{442887867}{2788056448} a^{7} + \frac{5}{32} a^{6} - \frac{171543165}{1394028224} a^{5} - \frac{5}{32} a^{4} + \frac{336072757}{697014112} a^{3} - \frac{3}{8} a^{2} - \frac{135882257}{348507056} a + \frac{1}{8}$

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:  $14$
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:  \( 1081704794.63 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T513:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 128 conjugacy class representatives for t20n513 are not computed
Character table for t20n513 is not computed

Intermediate fields

\(\Q(\sqrt{53}) \), 5.5.2382032.1, 10.10.300726051798272.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.18.44$x^{8} + 12 x^{6} + 4 x^{4} + 12$$4$$2$$18$$((C_8 : C_2):C_2):C_2$$[2, 3, 3, 7/2]^{4}$
53Data not computed