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Properties

Label 20.0.12096059838...2096.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{56}\cdot 11^{8}\cdot 23^{8}$
Root discriminant $63.70$
Ramified primes $2, 11, 23$
Class number $1176$ (GRH)
Class group $[1176]$ (GRH)
Galois group 20T277

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, 0, 23616, 0, 128208, 0, 246784, 0, 234718, 0, 125480, 0, 39716, 0, 7496, 0, 815, 0, 46, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 46*x^18 + 815*x^16 + 7496*x^14 + 39716*x^12 + 125480*x^10 + 234718*x^8 + 246784*x^6 + 128208*x^4 + 23616*x^2 + 64)
 
gp: K = bnfinit(x^20 + 46*x^18 + 815*x^16 + 7496*x^14 + 39716*x^12 + 125480*x^10 + 234718*x^8 + 246784*x^6 + 128208*x^4 + 23616*x^2 + 64, 1)
 

Normalized defining polynomial

\( x^{20} + 46 x^{18} + 815 x^{16} + 7496 x^{14} + 39716 x^{12} + 125480 x^{10} + 234718 x^{8} + 246784 x^{6} + 128208 x^{4} + 23616 x^{2} + 64 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1209605983843492469982030335989252096=2^{56}\cdot 11^{8}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.70$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{17} - \frac{1}{8} a^{14} + \frac{1}{16} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4064404132240} a^{18} - \frac{6130084315}{406440413224} a^{16} + \frac{41701126075}{812880826448} a^{14} + \frac{8630658076}{254025258265} a^{12} + \frac{11678265588}{50805051653} a^{10} - \frac{827970127}{5347900174} a^{8} + \frac{627744354579}{2032202066120} a^{6} + \frac{94320032976}{254025258265} a^{4} - \frac{192484491}{508050516530} a^{2} + \frac{19602753189}{254025258265}$, $\frac{1}{8128808264480} a^{19} - \frac{6130084315}{812880826448} a^{17} + \frac{41701126075}{1625761652896} a^{15} - \frac{219502625961}{2032202066120} a^{13} - \frac{4091989301}{406440413224} a^{11} - \frac{827970127}{10695800348} a^{9} + \frac{1643845387639}{4064404132240} a^{7} + \frac{442665324217}{1016101033060} a^{5} - \frac{63554435689}{254025258265} a^{3} - \frac{117211252538}{254025258265} a$

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

Class group and class number

$C_{1176}$, which has order $1176$ (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:  \( -1 \) (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:  \( 65638146.2726 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T277:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 48 conjugacy class representatives for t20n277
Character table for t20n277 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.5.16386304.1, 10.10.2148087670243328.1, 10.0.1099820887164583936.1, 10.0.137477610895572992.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.8.25.69$x^{8} + 6 x^{4} + 4 x^{2} + 30$$8$$1$$25$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 3, 7/2, 4, 17/4]^{2}$
2.8.25.69$x^{8} + 6 x^{4} + 4 x^{2} + 30$$8$$1$$25$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 3, 7/2, 4, 17/4]^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$23$23.3.2.1$x^{3} - 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
23.3.2.1$x^{3} - 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
23.3.2.1$x^{3} - 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
23.3.2.1$x^{3} - 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$