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Properties

Label 20.16.6584630745...0000.2
Degree $20$
Signature $[16, 2]$
Discriminant $2^{28}\cdot 5^{11}\cdot 3469^{5}$
Root discriminant $49.08$
Ramified primes $2, 5, 3469$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17345, 0, -154710, 0, 463389, 0, -570872, 0, 276498, 0, -28580, 0, -11774, 0, 2312, 0, 13, 0, -22, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 22*x^18 + 13*x^16 + 2312*x^14 - 11774*x^12 - 28580*x^10 + 276498*x^8 - 570872*x^6 + 463389*x^4 - 154710*x^2 + 17345)
 
gp: K = bnfinit(x^20 - 22*x^18 + 13*x^16 + 2312*x^14 - 11774*x^12 - 28580*x^10 + 276498*x^8 - 570872*x^6 + 463389*x^4 - 154710*x^2 + 17345, 1)
 

Normalized defining polynomial

\( x^{20} - 22 x^{18} + 13 x^{16} + 2312 x^{14} - 11774 x^{12} - 28580 x^{10} + 276498 x^{8} - 570872 x^{6} + 463389 x^{4} - 154710 x^{2} + 17345 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6584630745307392888012800000000000=2^{28}\cdot 5^{11}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.08$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{4} + \frac{1}{8}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{4} - \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{8} - \frac{1}{8} a^{6} + \frac{1}{8} a^{4} + \frac{1}{16} a^{2} - \frac{1}{16}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{16} a^{3} + \frac{3}{16} a + \frac{1}{4}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{8} - \frac{1}{16} a^{4} + \frac{1}{16}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} + \frac{1}{32} a^{9} - \frac{1}{32} a^{8} + \frac{3}{32} a^{5} - \frac{3}{32} a^{4} - \frac{5}{32} a + \frac{5}{32}$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{12} - \frac{1}{32} a^{10} + \frac{1}{32} a^{8} - \frac{1}{32} a^{6} + \frac{1}{32} a^{4} + \frac{1}{32} a^{2} - \frac{1}{32}$, $\frac{1}{64} a^{15} - \frac{1}{64} a^{14} - \frac{1}{64} a^{13} + \frac{1}{64} a^{12} + \frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{1}{64} a^{9} + \frac{1}{64} a^{8} - \frac{5}{64} a^{7} + \frac{5}{64} a^{6} + \frac{5}{64} a^{5} - \frac{5}{64} a^{4} + \frac{3}{64} a^{3} - \frac{3}{64} a^{2} - \frac{3}{64} a + \frac{3}{64}$, $\frac{1}{320} a^{16} - \frac{1}{40} a^{12} - \frac{1}{80} a^{10} + \frac{3}{160} a^{8} - \frac{3}{40} a^{6} - \frac{7}{40} a^{4} - \frac{5}{16} a^{2} - \frac{27}{64}$, $\frac{1}{320} a^{17} + \frac{1}{160} a^{13} - \frac{1}{32} a^{12} - \frac{1}{80} a^{11} + \frac{1}{20} a^{9} - \frac{1}{32} a^{8} - \frac{3}{40} a^{7} - \frac{13}{160} a^{5} - \frac{3}{32} a^{4} + \frac{3}{16} a^{3} - \frac{1}{2} a^{2} - \frac{5}{64} a - \frac{11}{32}$, $\frac{1}{1525084334532726400} a^{18} - \frac{938895807469421}{1525084334532726400} a^{16} + \frac{21183808191029}{190635541816590800} a^{14} - \frac{286982875137241}{95317770908295400} a^{12} - \frac{238048872560839}{30501686690654528} a^{10} - \frac{5243415932391269}{152508433453272640} a^{8} + \frac{94722870057057}{23829442727073850} a^{6} - \frac{30782980321774823}{190635541816590800} a^{4} - \frac{78532180554049487}{305016866906545280} a^{2} - \frac{69863401930208069}{305016866906545280}$, $\frac{1}{1525084334532726400} a^{19} - \frac{938895807469421}{1525084334532726400} a^{17} + \frac{21183808191029}{190635541816590800} a^{15} - \frac{286982875137241}{95317770908295400} a^{13} - \frac{238048872560839}{30501686690654528} a^{11} - \frac{5243415932391269}{152508433453272640} a^{9} + \frac{94722870057057}{23829442727073850} a^{7} + \frac{16875905132372877}{190635541816590800} a^{5} - \frac{1}{4} a^{4} + \frac{73976252899223153}{305016866906545280} a^{3} - \frac{1}{2} a^{2} + \frac{6390814796428251}{305016866906545280} a - \frac{1}{4}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $17$
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:  \( 8454796039.47 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 $20$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3469Data not computed