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Properties

Label 20.10.1698519628...6331.3
Degree $20$
Signature $[10, 5]$
Discriminant $-\,11^{18}\cdot 1451^{3}$
Root discriminant $25.79$
Ramified primes $11, 1451$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T432

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 17, 56, -72, -764, 1236, 771, -2631, 1493, 591, -1508, 1314, -639, 28, 302, -345, 230, -108, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 36*x^18 - 108*x^17 + 230*x^16 - 345*x^15 + 302*x^14 + 28*x^13 - 639*x^12 + 1314*x^11 - 1508*x^10 + 591*x^9 + 1493*x^8 - 2631*x^7 + 771*x^6 + 1236*x^5 - 764*x^4 - 72*x^3 + 56*x^2 + 17*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 36*x^18 - 108*x^17 + 230*x^16 - 345*x^15 + 302*x^14 + 28*x^13 - 639*x^12 + 1314*x^11 - 1508*x^10 + 591*x^9 + 1493*x^8 - 2631*x^7 + 771*x^6 + 1236*x^5 - 764*x^4 - 72*x^3 + 56*x^2 + 17*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 36 x^{18} - 108 x^{17} + 230 x^{16} - 345 x^{15} + 302 x^{14} + 28 x^{13} - 639 x^{12} + 1314 x^{11} - 1508 x^{10} + 591 x^{9} + 1493 x^{8} - 2631 x^{7} + 771 x^{6} + 1236 x^{5} - 764 x^{4} - 72 x^{3} + 56 x^{2} + 17 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-16985196289500337453529206331=-\,11^{18}\cdot 1451^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.79$
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:  $11, 1451$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2676213531961892187077} a^{19} + \frac{41437593161458033965}{2676213531961892187077} a^{18} + \frac{1336436116656326655059}{2676213531961892187077} a^{17} + \frac{195640686107178631947}{2676213531961892187077} a^{16} - \frac{635564733468076975462}{2676213531961892187077} a^{15} + \frac{292137902027132524275}{2676213531961892187077} a^{14} - \frac{1012642115855048841103}{2676213531961892187077} a^{13} - \frac{723422686081095640010}{2676213531961892187077} a^{12} - \frac{1298264027366438543477}{2676213531961892187077} a^{11} + \frac{1263014221658246167944}{2676213531961892187077} a^{10} + \frac{976005622521532499051}{2676213531961892187077} a^{9} + \frac{876246223212847731443}{2676213531961892187077} a^{8} + \frac{1220330113289663814319}{2676213531961892187077} a^{7} + \frac{805391655576797869471}{2676213531961892187077} a^{6} - \frac{1095407378443907765386}{2676213531961892187077} a^{5} - \frac{260823537727542342736}{2676213531961892187077} a^{4} - \frac{528850600574228017387}{2676213531961892187077} a^{3} - \frac{1075302075924099474566}{2676213531961892187077} a^{2} - \frac{775899227268586393853}{2676213531961892187077} a + \frac{909532883930936150064}{2676213531961892187077}$

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:  $14$
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:  \( 4686457.89326 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T432:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n432 are not computed
Character table for t20n432 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.3421382099641.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ $20$

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
11Data not computed
1451Data not computed