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Properties

Label 20.0.54104064570...5232.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{37}\cdot 89^{8}$
Root discriminant $21.71$
Ramified primes $2, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T803

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, 0, -80, 0, 76, 0, -100, 0, 249, 0, -368, 0, 304, 0, -152, 0, 49, 0, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 + 49*x^16 - 152*x^14 + 304*x^12 - 368*x^10 + 249*x^8 - 100*x^6 + 76*x^4 - 80*x^2 + 32)
 
gp: K = bnfinit(x^20 - 10*x^18 + 49*x^16 - 152*x^14 + 304*x^12 - 368*x^10 + 249*x^8 - 100*x^6 + 76*x^4 - 80*x^2 + 32, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{18} + 49 x^{16} - 152 x^{14} + 304 x^{12} - 368 x^{10} + 249 x^{8} - 100 x^{6} + 76 x^{4} - 80 x^{2} + 32 \)

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:  \(541040645705284358852575232=2^{37}\cdot 89^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.71$
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:  $2, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{588} a^{16} - \frac{2}{147} a^{14} - \frac{83}{588} a^{12} + \frac{11}{294} a^{10} - \frac{5}{147} a^{8} - \frac{5}{147} a^{6} + \frac{59}{196} a^{4} + \frac{37}{98} a^{2} - \frac{5}{147}$, $\frac{1}{1176} a^{17} - \frac{1}{147} a^{15} + \frac{505}{1176} a^{13} - \frac{283}{588} a^{11} - \frac{5}{294} a^{9} + \frac{71}{147} a^{7} + \frac{59}{392} a^{5} + \frac{37}{196} a^{3} + \frac{71}{147} a$, $\frac{1}{2352} a^{18} - \frac{1}{4} a^{15} + \frac{3}{16} a^{14} - \frac{1}{2} a^{13} + \frac{187}{392} a^{12} - \frac{1}{4} a^{11} + \frac{13}{196} a^{10} + \frac{17}{98} a^{8} - \frac{1159}{2352} a^{6} - \frac{17}{56} a^{4} - \frac{1}{4} a^{3} + \frac{73}{147} a^{2} - \frac{10}{147}$, $\frac{1}{4704} a^{19} - \frac{1}{1176} a^{16} - \frac{5}{32} a^{15} - \frac{143}{588} a^{14} + \frac{187}{784} a^{13} + \frac{83}{1176} a^{12} + \frac{111}{392} a^{11} + \frac{34}{147} a^{10} - \frac{81}{196} a^{9} + \frac{5}{294} a^{8} + \frac{1193}{4704} a^{7} - \frac{71}{147} a^{6} - \frac{17}{112} a^{5} - \frac{59}{392} a^{4} - \frac{1}{588} a^{3} - \frac{43}{98} a^{2} + \frac{137}{294} a - \frac{71}{147}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{99}{196} a^{18} - \frac{891}{196} a^{16} + \frac{3943}{196} a^{14} - \frac{1567}{28} a^{12} + \frac{1327}{14} a^{10} - 84 a^{8} + \frac{6175}{196} a^{6} - \frac{2405}{196} a^{4} + \frac{2329}{98} a^{2} - \frac{681}{49} \) (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:  \( 1582244.79704 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T803:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 126 conjugacy class representatives for t20n803 are not computed
Character table for t20n803 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.3.31684.1, 10.0.64248054784.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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$2.2.2.1$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}$$[2, 3, 7/2]$
2.12.24.96$x^{12} + 12 x^{11} - 10 x^{10} + 10 x^{8} + 4 x^{6} + 16 x^{4} + 16 x^{3} + 16 x^{2} + 16 x + 8$$4$$3$$24$12T87$[2, 2, 2, 3, 3, 3]^{3}$
89Data not computed