↑

Properties

Label 20.0.27956717291...4752.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 53^{13}$
Root discriminant $37.35$
Ramified primes $2, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2:F_5$ (as 20T19)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![212, 0, -720, 0, 3429, 0, -4878, 0, 8229, 0, -5960, 0, 1830, 0, -246, 0, 48, 0, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 12*x^18 + 48*x^16 - 246*x^14 + 1830*x^12 - 5960*x^10 + 8229*x^8 - 4878*x^6 + 3429*x^4 - 720*x^2 + 212)
 
gp: K = bnfinit(x^20 - 12*x^18 + 48*x^16 - 246*x^14 + 1830*x^12 - 5960*x^10 + 8229*x^8 - 4878*x^6 + 3429*x^4 - 720*x^2 + 212, 1)
 

Normalized defining polynomial

\( x^{20} - 12 x^{18} + 48 x^{16} - 246 x^{14} + 1830 x^{12} - 5960 x^{10} + 8229 x^{8} - 4878 x^{6} + 3429 x^{4} - 720 x^{2} + 212 \)

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:  \(27956717291381500794294870474752=2^{30}\cdot 53^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.35$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2932} a^{14} - \frac{77}{1466} a^{12} - \frac{1}{4} a^{11} + \frac{53}{2932} a^{10} + \frac{127}{2932} a^{8} + \frac{355}{2932} a^{6} - \frac{1}{4} a^{5} + \frac{1023}{2932} a^{4} - \frac{1}{4} a^{3} - \frac{169}{2932} a^{2} - \frac{345}{1466}$, $\frac{1}{2932} a^{15} - \frac{77}{1466} a^{13} - \frac{1}{4} a^{12} + \frac{53}{2932} a^{11} + \frac{127}{2932} a^{9} + \frac{355}{2932} a^{7} - \frac{1}{4} a^{6} + \frac{1023}{2932} a^{5} - \frac{1}{4} a^{4} - \frac{169}{2932} a^{3} - \frac{345}{1466} a$, $\frac{1}{2932} a^{16} - \frac{1}{4} a^{13} - \frac{207}{2932} a^{12} - \frac{507}{2932} a^{10} - \frac{611}{2932} a^{8} - \frac{1}{4} a^{7} - \frac{15}{2932} a^{6} - \frac{1}{4} a^{5} - \frac{955}{2932} a^{4} - \frac{82}{733} a^{2} - \frac{177}{733}$, $\frac{1}{2932} a^{17} - \frac{207}{2932} a^{13} + \frac{113}{1466} a^{11} - \frac{1}{4} a^{10} - \frac{611}{2932} a^{9} - \frac{1}{2} a^{8} - \frac{15}{2932} a^{7} - \frac{111}{1466} a^{5} - \frac{1}{4} a^{4} + \frac{405}{2932} a^{3} + \frac{1}{4} a^{2} - \frac{177}{733} a - \frac{1}{2}$, $\frac{1}{49713528932} a^{18} + \frac{1043879}{12428382233} a^{16} - \frac{1640283}{49713528932} a^{14} + \frac{3980237871}{24856764466} a^{12} - \frac{1}{4} a^{11} + \frac{2707873371}{49713528932} a^{10} - \frac{1}{2} a^{9} - \frac{3568187271}{49713528932} a^{8} - \frac{218760315}{24856764466} a^{6} - \frac{1}{4} a^{5} - \frac{10050633069}{49713528932} a^{4} + \frac{1}{4} a^{3} + \frac{9486729415}{24856764466} a^{2} - \frac{1}{2} a - \frac{47766670}{12428382233}$, $\frac{1}{49713528932} a^{19} + \frac{1043879}{12428382233} a^{17} - \frac{1640283}{49713528932} a^{15} + \frac{3980237871}{24856764466} a^{13} - \frac{1}{4} a^{12} + \frac{2707873371}{49713528932} a^{11} - \frac{3568187271}{49713528932} a^{9} - \frac{218760315}{24856764466} a^{7} - \frac{1}{4} a^{6} - \frac{10050633069}{49713528932} a^{5} - \frac{1}{4} a^{4} + \frac{9486729415}{24856764466} a^{3} - \frac{47766670}{12428382233} a$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{2458}{812897} a^{18} - \frac{29555}{812897} a^{16} + \frac{237145}{1625794} a^{14} - \frac{604856}{812897} a^{12} + \frac{8994829}{1625794} a^{10} - \frac{29414491}{1625794} a^{8} + \frac{40301377}{1625794} a^{6} - \frac{20860769}{1625794} a^{4} + \frac{11077117}{1625794} a^{2} - \frac{831656}{812897} \) (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:  \( 75297483.0313 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:F_5$ (as 20T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 14 conjugacy class representatives for $C_2^2:F_5$
Character table for $C_2^2:F_5$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.3392.1, 5.5.2382032.1, 10.0.22696305796096.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$53$53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$