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Properties

Label 20.20.9683254139...7744.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 11^{8}\cdot 29^{10}$
Root discriminant $39.75$
Ramified primes $2, 11, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{10}\times D_5$ (as 20T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -152, 28, 2218, -2548, -7566, 13328, 6610, -24144, 6604, 16028, -11747, -1552, 3972, -844, -398, 176, -2, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 2*x^18 + 176*x^17 - 398*x^16 - 844*x^15 + 3972*x^14 - 1552*x^13 - 11747*x^12 + 16028*x^11 + 6604*x^10 - 24144*x^9 + 6610*x^8 + 13328*x^7 - 7566*x^6 - 2548*x^5 + 2218*x^4 + 28*x^3 - 152*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 - 2*x^18 + 176*x^17 - 398*x^16 - 844*x^15 + 3972*x^14 - 1552*x^13 - 11747*x^12 + 16028*x^11 + 6604*x^10 - 24144*x^9 + 6610*x^8 + 13328*x^7 - 7566*x^6 - 2548*x^5 + 2218*x^4 + 28*x^3 - 152*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} - 2 x^{18} + 176 x^{17} - 398 x^{16} - 844 x^{15} + 3972 x^{14} - 1552 x^{13} - 11747 x^{12} + 16028 x^{11} + 6604 x^{10} - 24144 x^{9} + 6610 x^{8} + 13328 x^{7} - 7566 x^{6} - 2548 x^{5} + 2218 x^{4} + 28 x^{3} - 152 x^{2} - 4 x + 1 \)

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:  \(96832541395306793661914618527744=2^{30}\cdot 11^{8}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.75$
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:  $2, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{6} a^{16} + \frac{1}{6} a^{15} + \frac{1}{6} a^{14} - \frac{1}{6} a^{13} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{2} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{17} + \frac{1}{6} a^{14} + \frac{1}{6} a^{13} + \frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{3} a^{9} - \frac{1}{2} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{6} a^{4} + \frac{1}{6} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{18} + \frac{1}{6} a^{15} + \frac{1}{6} a^{14} + \frac{1}{6} a^{13} + \frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{530970633975246} a^{19} - \frac{39621732798587}{530970633975246} a^{18} - \frac{8361262064056}{265485316987623} a^{17} + \frac{34398884723735}{530970633975246} a^{16} - \frac{44154098716149}{176990211325082} a^{15} + \frac{50760665918534}{265485316987623} a^{14} - \frac{18933959792006}{265485316987623} a^{13} + \frac{918376366687}{88495105662541} a^{12} - \frac{1260982792000}{265485316987623} a^{11} + \frac{33763652555795}{265485316987623} a^{10} + \frac{97926326912147}{530970633975246} a^{9} - \frac{30189152668087}{88495105662541} a^{8} - \frac{34818804601685}{88495105662541} a^{7} + \frac{103726644993137}{265485316987623} a^{6} - \frac{27335526872926}{265485316987623} a^{5} - \frac{114947421021781}{530970633975246} a^{4} - \frac{125035034589112}{265485316987623} a^{3} + \frac{57224715679965}{176990211325082} a^{2} - \frac{11048301366575}{88495105662541} a - \frac{43172831652173}{176990211325082}$

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:  \( 4306703261.49 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{10}\times D_5$ (as 20T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 40 conjugacy class representatives for $C_{10}\times D_5$
Character table for $C_{10}\times D_5$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{58}) \), \(\Q(\sqrt{2}, \sqrt{29})\), 10.10.9840352706854912.1

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

Sibling fields

Degree 20 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
29Data not computed