↑

Properties

Label 20.0.70664371030...0656.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 7^{10}\cdot 13^{12}$
Root discriminant $34.87$
Ramified primes $2, 7, 13$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4487, -10388, -2294, 7172, 70862, -131568, 44046, 72952, -82312, 32620, -8, -9624, 9614, -6148, 2946, -1236, 462, -140, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 36*x^18 - 140*x^17 + 462*x^16 - 1236*x^15 + 2946*x^14 - 6148*x^13 + 9614*x^12 - 9624*x^11 - 8*x^10 + 32620*x^9 - 82312*x^8 + 72952*x^7 + 44046*x^6 - 131568*x^5 + 70862*x^4 + 7172*x^3 - 2294*x^2 - 10388*x + 4487)
 
gp: K = bnfinit(x^20 - 8*x^19 + 36*x^18 - 140*x^17 + 462*x^16 - 1236*x^15 + 2946*x^14 - 6148*x^13 + 9614*x^12 - 9624*x^11 - 8*x^10 + 32620*x^9 - 82312*x^8 + 72952*x^7 + 44046*x^6 - 131568*x^5 + 70862*x^4 + 7172*x^3 - 2294*x^2 - 10388*x + 4487, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 36 x^{18} - 140 x^{17} + 462 x^{16} - 1236 x^{15} + 2946 x^{14} - 6148 x^{13} + 9614 x^{12} - 9624 x^{11} - 8 x^{10} + 32620 x^{9} - 82312 x^{8} + 72952 x^{7} + 44046 x^{6} - 131568 x^{5} + 70862 x^{4} + 7172 x^{3} - 2294 x^{2} - 10388 x + 4487 \)

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:  \(7066437103077000852134120390656=2^{30}\cdot 7^{10}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.87$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{7}$, $\frac{1}{8} a^{18} + \frac{1}{8} a^{16} - \frac{1}{8} a^{14} + \frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{2} a^{7} + \frac{3}{8} a^{6} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{150594461878087884468413502610997670195328} a^{19} + \frac{5220519520317408235400472158640941579975}{150594461878087884468413502610997670195328} a^{18} + \frac{30638364335653906941685443966467523143405}{150594461878087884468413502610997670195328} a^{17} - \frac{1742418223613754800851993245844712444649}{150594461878087884468413502610997670195328} a^{16} + \frac{21745438350729423153577261753075500857543}{150594461878087884468413502610997670195328} a^{15} + \frac{26971831557195326418124530984907555836693}{150594461878087884468413502610997670195328} a^{14} + \frac{20566936224443157439729100354782606477981}{150594461878087884468413502610997670195328} a^{13} + \frac{18768280718171418541445681796975985299759}{150594461878087884468413502610997670195328} a^{12} - \frac{4789460840193356794551579295332393830097}{150594461878087884468413502610997670195328} a^{11} + \frac{37432472685429766285195723136291897389929}{150594461878087884468413502610997670195328} a^{10} - \frac{35384073596961919576120677782283481498625}{150594461878087884468413502610997670195328} a^{9} - \frac{24888530825258517525013684929395322803043}{150594461878087884468413502610997670195328} a^{8} + \frac{36284854840896808164827299865626561091339}{150594461878087884468413502610997670195328} a^{7} - \frac{7435336650646412491901138319304016005539}{150594461878087884468413502610997670195328} a^{6} + \frac{50910561078543473669213147030568372482145}{150594461878087884468413502610997670195328} a^{5} - \frac{46875999299158030394674183045314709151361}{150594461878087884468413502610997670195328} a^{4} + \frac{4031725710678433767467225623039111944031}{150594461878087884468413502610997670195328} a^{3} + \frac{67085892447474943542954819963981055971349}{150594461878087884468413502610997670195328} a^{2} - \frac{61699007275397959409669120734210489697563}{150594461878087884468413502610997670195328} a + \frac{71900752968812765664043021905388832918679}{150594461878087884468413502610997670195328}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 9631410.14359 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{-14}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{-7})\), 5.5.6889792.1, 10.0.2658277092982784.1, 10.0.332284636622848.1, 10.10.379753870426112.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$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]$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$