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Properties

Label 20.4.28951655329...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{58}\cdot 5^{12}\cdot 7^{6}\cdot 769^{4}$
Root discriminant $132.77$
Ramified primes $2, 5, 7, 769$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T872

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![470596, 0, 2151296, 0, 2324168, 0, 864360, 0, -23716, 0, -94388, 0, -19878, 0, 140, 0, 456, 0, 44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 44*x^18 + 456*x^16 + 140*x^14 - 19878*x^12 - 94388*x^10 - 23716*x^8 + 864360*x^6 + 2324168*x^4 + 2151296*x^2 + 470596)
 
gp: K = bnfinit(x^20 + 44*x^18 + 456*x^16 + 140*x^14 - 19878*x^12 - 94388*x^10 - 23716*x^8 + 864360*x^6 + 2324168*x^4 + 2151296*x^2 + 470596, 1)
 

Normalized defining polynomial

\( x^{20} + 44 x^{18} + 456 x^{16} + 140 x^{14} - 19878 x^{12} - 94388 x^{10} - 23716 x^{8} + 864360 x^{6} + 2324168 x^{4} + 2151296 x^{2} + 470596 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2895165532916258147798358163456000000000000=2^{58}\cdot 5^{12}\cdot 7^{6}\cdot 769^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $132.77$
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:  $2, 5, 7, 769$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{14} a^{10} + \frac{1}{7} a^{8} - \frac{3}{7} a^{6} + \frac{1}{7} a^{2}$, $\frac{1}{14} a^{11} + \frac{1}{7} a^{9} - \frac{3}{7} a^{7} + \frac{1}{7} a^{3}$, $\frac{1}{98} a^{12} + \frac{1}{49} a^{10} - \frac{10}{49} a^{8} + \frac{8}{49} a^{4}$, $\frac{1}{98} a^{13} + \frac{1}{49} a^{11} - \frac{10}{49} a^{9} + \frac{8}{49} a^{5}$, $\frac{1}{686} a^{14} + \frac{1}{343} a^{12} - \frac{10}{343} a^{10} + \frac{2}{7} a^{8} + \frac{155}{343} a^{6} + \frac{3}{7} a^{4} + \frac{2}{7} a^{2}$, $\frac{1}{686} a^{15} + \frac{1}{343} a^{13} - \frac{10}{343} a^{11} + \frac{2}{7} a^{9} + \frac{155}{343} a^{7} + \frac{3}{7} a^{5} + \frac{2}{7} a^{3}$, $\frac{1}{4802} a^{16} + \frac{1}{2401} a^{14} - \frac{10}{2401} a^{12} - \frac{3}{98} a^{10} - \frac{874}{2401} a^{8} + \frac{3}{49} a^{6} + \frac{9}{49} a^{4} + \frac{1}{7} a^{2}$, $\frac{1}{4802} a^{17} + \frac{1}{2401} a^{15} - \frac{10}{2401} a^{13} - \frac{3}{98} a^{11} - \frac{874}{2401} a^{9} + \frac{3}{49} a^{7} + \frac{9}{49} a^{5} + \frac{1}{7} a^{3}$, $\frac{1}{240914374517390} a^{18} + \frac{10513554}{797729716945} a^{16} + \frac{74204241444}{120457187258695} a^{14} + \frac{82923938988}{17208169608385} a^{12} + \frac{1717311247399}{48182874903478} a^{10} - \frac{6070696924932}{17208169608385} a^{8} - \frac{1122511955926}{2458309944055} a^{6} + \frac{21857756194}{50169590695} a^{4} + \frac{11731189}{10033918139} a^{2} - \frac{542399131}{7167084385}$, $\frac{1}{481828749034780} a^{19} + \frac{5256777}{797729716945} a^{17} + \frac{37102120722}{120457187258695} a^{15} + \frac{41461969494}{17208169608385} a^{13} - \frac{862161337139}{48182874903478} a^{11} - \frac{8529006868987}{34416339216770} a^{9} + \frac{1194679696362}{2458309944055} a^{7} + \frac{10928878097}{50169590695} a^{5} - \frac{710842844}{10033918139} a^{3} + \frac{3312342627}{7167084385} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 25001806619100 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T872:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 204800
The 116 conjugacy class representatives for t20n872 are not computed
Character table for t20n872 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.4844429312000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
2Data not computed
5Data not computed
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.2$x^{8} + 49 x^{4} - 1029 x^{2} + 12005$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
769Data not computed