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Properties

Label 20.0.69313403908...3312.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 7^{15}\cdot 11^{9}$
Root discriminant $43.86$
Ramified primes $2, 3, 7, 11$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10916801973, 0, 0, 0, 346565142, 0, 0, 0, 2750517, 0, -166698, 0, 31752, 0, 2646, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 2646*x^14 + 31752*x^12 - 166698*x^10 + 2750517*x^8 + 346565142*x^4 + 10916801973)
 
gp: K = bnfinit(x^20 + 2646*x^14 + 31752*x^12 - 166698*x^10 + 2750517*x^8 + 346565142*x^4 + 10916801973, 1)
 

Normalized defining polynomial

\( x^{20} + 2646 x^{14} + 31752 x^{12} - 166698 x^{10} + 2750517 x^{8} + 346565142 x^{4} + 10916801973 \)

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:  \(693134039080601839001252435853312=2^{20}\cdot 3^{10}\cdot 7^{15}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.86$
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:  $2, 3, 7, 11$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{63} a^{4}$, $\frac{1}{63} a^{5}$, $\frac{1}{189} a^{6}$, $\frac{1}{189} a^{7}$, $\frac{1}{3969} a^{8}$, $\frac{1}{3969} a^{9}$, $\frac{1}{47628} a^{10} + \frac{1}{252} a^{4} - \frac{1}{6} a^{2} - \frac{1}{4}$, $\frac{1}{47628} a^{11} + \frac{1}{252} a^{5} - \frac{1}{6} a^{3} - \frac{1}{4} a$, $\frac{1}{1000188} a^{12} - \frac{1}{756} a^{6} - \frac{1}{126} a^{4} + \frac{1}{12} a^{2}$, $\frac{1}{1000188} a^{13} - \frac{1}{756} a^{7} - \frac{1}{126} a^{5} + \frac{1}{12} a^{3}$, $\frac{1}{3000564} a^{14} + \frac{1}{15876} a^{8} - \frac{1}{378} a^{6} - \frac{1}{252} a^{4}$, $\frac{1}{3000564} a^{15} + \frac{1}{15876} a^{9} - \frac{1}{378} a^{7} - \frac{1}{252} a^{5}$, $\frac{1}{63011844} a^{16} - \frac{1}{7938} a^{8} + \frac{1}{756} a^{6} + \frac{1}{252} a^{4} - \frac{1}{6} a^{2} - \frac{1}{4}$, $\frac{1}{63011844} a^{17} - \frac{1}{7938} a^{9} + \frac{1}{756} a^{7} + \frac{1}{252} a^{5} - \frac{1}{6} a^{3} - \frac{1}{4} a$, $\frac{1}{547577902295676} a^{18} - \frac{643105}{91262983715946} a^{16} + \frac{15493}{321915286476} a^{14} - \frac{1136783}{2897237578284} a^{12} + \frac{70447}{9854549586} a^{10} - \frac{1758431}{22993949034} a^{8} + \frac{303512}{182491659} a^{6} + \frac{1748983}{729966636} a^{4} - \frac{386215}{8690079} a^{2} + \frac{2321853}{5793386}$, $\frac{1}{547577902295676} a^{19} - \frac{643105}{91262983715946} a^{17} + \frac{15493}{321915286476} a^{15} - \frac{1136783}{2897237578284} a^{13} + \frac{70447}{9854549586} a^{11} - \frac{1758431}{22993949034} a^{9} + \frac{303512}{182491659} a^{7} + \frac{1748983}{729966636} a^{5} - \frac{386215}{8690079} a^{3} + \frac{2321853}{5793386} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( 15627805.568647645 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.543312.1, 10.0.246071287.1

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 R $20$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{10}$ $20$

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$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
3Data not computed
7Data not computed
$11$11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$