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Properties

Label 20.20.1465454101...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 5^{34}\cdot 7^{4}$
Root discriminant $45.53$
Ramified primes $2, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:C_5$ (as 20T44)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2401, 0, -36015, 0, 139160, 0, -228585, 0, 198790, 0, -100872, 0, 31140, 0, -5885, 0, 660, 0, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 40*x^18 + 660*x^16 - 5885*x^14 + 31140*x^12 - 100872*x^10 + 198790*x^8 - 228585*x^6 + 139160*x^4 - 36015*x^2 + 2401)
 
gp: K = bnfinit(x^20 - 40*x^18 + 660*x^16 - 5885*x^14 + 31140*x^12 - 100872*x^10 + 198790*x^8 - 228585*x^6 + 139160*x^4 - 36015*x^2 + 2401, 1)
 

Normalized defining polynomial

\( x^{20} - 40 x^{18} + 660 x^{16} - 5885 x^{14} + 31140 x^{12} - 100872 x^{10} + 198790 x^{8} - 228585 x^{6} + 139160 x^{4} - 36015 x^{2} + 2401 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1465454101562500000000000000000000=2^{20}\cdot 5^{34}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.53$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} - \frac{2}{7} a^{8} + \frac{2}{7} a^{6} + \frac{3}{7} a^{4} - \frac{3}{7} a^{2}$, $\frac{1}{7} a^{11} - \frac{2}{7} a^{9} + \frac{2}{7} a^{7} + \frac{3}{7} a^{5} - \frac{3}{7} a^{3}$, $\frac{1}{7} a^{12} - \frac{2}{7} a^{8} + \frac{3}{7} a^{4} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{13} - \frac{2}{7} a^{9} + \frac{3}{7} a^{5} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{14} + \frac{3}{7} a^{8} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{15} + \frac{3}{7} a^{9} + \frac{1}{7} a^{3}$, $\frac{1}{49} a^{16} + \frac{2}{49} a^{14} + \frac{2}{49} a^{12} + \frac{2}{49} a^{10} - \frac{17}{49} a^{8} - \frac{16}{49} a^{6} + \frac{4}{49} a^{4}$, $\frac{1}{49} a^{17} + \frac{2}{49} a^{15} + \frac{2}{49} a^{13} + \frac{2}{49} a^{11} - \frac{17}{49} a^{9} - \frac{16}{49} a^{7} + \frac{4}{49} a^{5}$, $\frac{1}{84117251743} a^{18} + \frac{536866563}{84117251743} a^{16} - \frac{4324501270}{84117251743} a^{14} + \frac{3169873380}{84117251743} a^{12} - \frac{2662101053}{84117251743} a^{10} + \frac{19182237002}{84117251743} a^{8} + \frac{3060681873}{84117251743} a^{6} - \frac{3349422292}{12016750249} a^{4} + \frac{131448936}{1716678607} a^{2} - \frac{80588730}{245239801}$, $\frac{1}{84117251743} a^{19} + \frac{536866563}{84117251743} a^{17} - \frac{4324501270}{84117251743} a^{15} + \frac{3169873380}{84117251743} a^{13} - \frac{2662101053}{84117251743} a^{11} + \frac{19182237002}{84117251743} a^{9} + \frac{3060681873}{84117251743} a^{7} - \frac{3349422292}{12016750249} a^{5} + \frac{131448936}{1716678607} a^{3} - \frac{80588730}{245239801} a$

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:  $19$
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:  \( 16358901541.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T44):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 16 conjugacy class representatives for $C_2\times C_2^4:C_5$
Character table for $C_2\times C_2^4:C_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.390625.1, \(\Q(\zeta_{25})^+\), 10.10.38281250000000000.2, 10.10.7656250000000000.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$5$5.10.17.1$x^{10} - 5 x^{8} + 5$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.17.1$x^{10} - 5 x^{8} + 5$$10$$1$$17$$C_{10}$$[2]_{2}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$