↑

Properties

Label 20.0.32024912144...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 17^{2}\cdot 641^{5}$
Root discriminant $29.87$
Ramified primes $2, 5, 17, 641$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T756

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![185249, 0, -47345, 0, -4421, 0, 39229, 0, -13868, 0, 3253, 0, 564, 0, -139, 0, 24, 0, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 2*x^18 + 24*x^16 - 139*x^14 + 564*x^12 + 3253*x^10 - 13868*x^8 + 39229*x^6 - 4421*x^4 - 47345*x^2 + 185249)
 
gp: K = bnfinit(x^20 + 2*x^18 + 24*x^16 - 139*x^14 + 564*x^12 + 3253*x^10 - 13868*x^8 + 39229*x^6 - 4421*x^4 - 47345*x^2 + 185249, 1)
 

Normalized defining polynomial

\( x^{20} + 2 x^{18} + 24 x^{16} - 139 x^{14} + 564 x^{12} + 3253 x^{10} - 13868 x^{8} + 39229 x^{6} - 4421 x^{4} - 47345 x^{2} + 185249 \)

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:  \(320249121440041871360000000000=2^{20}\cdot 5^{10}\cdot 17^{2}\cdot 641^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.87$
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:  $2, 5, 17, 641$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{278791691441754417114314373} a^{18} + \frac{18821953745966530089807418}{92930563813918139038104791} a^{16} - \frac{30689928380607742745081922}{92930563813918139038104791} a^{14} + \frac{105623880650502142057887830}{278791691441754417114314373} a^{12} + \frac{18906704696347331788873310}{278791691441754417114314373} a^{10} + \frac{1211955683467972382348064}{92930563813918139038104791} a^{8} + \frac{119214706114783991420316814}{278791691441754417114314373} a^{6} + \frac{100456641309734935246998857}{278791691441754417114314373} a^{4} - \frac{46052472145644480854816980}{92930563813918139038104791} a^{2} - \frac{5851326235794072955264951}{16399511261279671594959669}$, $\frac{1}{278791691441754417114314373} a^{19} + \frac{18821953745966530089807418}{92930563813918139038104791} a^{17} - \frac{30689928380607742745081922}{92930563813918139038104791} a^{15} + \frac{105623880650502142057887830}{278791691441754417114314373} a^{13} + \frac{18906704696347331788873310}{278791691441754417114314373} a^{11} + \frac{1211955683467972382348064}{92930563813918139038104791} a^{9} + \frac{119214706114783991420316814}{278791691441754417114314373} a^{7} + \frac{100456641309734935246998857}{278791691441754417114314373} a^{5} - \frac{46052472145644480854816980}{92930563813918139038104791} a^{3} - \frac{5851326235794072955264951}{16399511261279671594959669} a$

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:  \( -1 \) (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:  \( 1459379.53638 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T756:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n756 are not computed
Character table for t20n756 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.2.1284003125.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$17$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
641Data not computed