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Properties

Label 20.8.86945198460...9381.1
Degree $20$
Signature $[8, 6]$
Discriminant $11^{16}\cdot 5741^{3}$
Root discriminant $24.94$
Ramified primes $11, 5741$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T335

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 4, -3, -47, -49, -95, -127, 60, 78, 214, 290, -11, -85, 20, 18, 3, -4, -7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^18 - 4*x^17 + 3*x^16 + 18*x^15 + 20*x^14 - 85*x^13 - 11*x^12 + 290*x^11 + 214*x^10 + 78*x^9 + 60*x^8 - 127*x^7 - 95*x^6 - 49*x^5 - 47*x^4 - 3*x^3 + 4*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^20 - 7*x^18 - 4*x^17 + 3*x^16 + 18*x^15 + 20*x^14 - 85*x^13 - 11*x^12 + 290*x^11 + 214*x^10 + 78*x^9 + 60*x^8 - 127*x^7 - 95*x^6 - 49*x^5 - 47*x^4 - 3*x^3 + 4*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{18} - 4 x^{17} + 3 x^{16} + 18 x^{15} + 20 x^{14} - 85 x^{13} - 11 x^{12} + 290 x^{11} + 214 x^{10} + 78 x^{9} + 60 x^{8} - 127 x^{7} - 95 x^{6} - 49 x^{5} - 47 x^{4} - 3 x^{3} + 4 x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8694519846067650027294539381=11^{16}\cdot 5741^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.94$
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:  $11, 5741$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1658991996198336951427} a^{19} - \frac{252952529977120522756}{1658991996198336951427} a^{18} - \frac{740638554398655628170}{1658991996198336951427} a^{17} + \frac{221415509657421949988}{1658991996198336951427} a^{16} - \frac{281649150779497164215}{1658991996198336951427} a^{15} + \frac{511497830425300909706}{1658991996198336951427} a^{14} + \frac{152214040869141999712}{1658991996198336951427} a^{13} - \frac{819324459691849896465}{1658991996198336951427} a^{12} + \frac{27827928642754627721}{1658991996198336951427} a^{11} - \frac{547338055456922250982}{1658991996198336951427} a^{10} + \frac{737483566280108664408}{1658991996198336951427} a^{9} - \frac{727960160836933382073}{1658991996198336951427} a^{8} - \frac{537254432193421880345}{1658991996198336951427} a^{7} - \frac{120275427087931298425}{1658991996198336951427} a^{6} + \frac{534289431511585974394}{1658991996198336951427} a^{5} - \frac{782723384718553725729}{1658991996198336951427} a^{4} + \frac{305052654066320589532}{1658991996198336951427} a^{3} + \frac{822463362833430926574}{1658991996198336951427} a^{2} + \frac{447823914135665256695}{1658991996198336951427} a - \frac{674999296270612872440}{1658991996198336951427}$

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:  $13$
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:  \( 2286594.01726 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T335:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 224 conjugacy class representatives for t20n335 are not computed
Character table for t20n335 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.1230634335821.1

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 $20$ $20$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ $20$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
5741Data not computed