↑

Properties

Label 20.4.45215251470...5625.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{10}\cdot 5^{10}\cdot 601^{5}$
Root discriminant $19.18$
Ramified primes $3, 5, 601$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times D_5\wr C_2$ (as 20T92)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11, -17, 118, 51, -391, 420, -264, 25, -548, 893, -779, 394, -395, 306, -62, 46, -23, -3, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^17 - 23*x^16 + 46*x^15 - 62*x^14 + 306*x^13 - 395*x^12 + 394*x^11 - 779*x^10 + 893*x^9 - 548*x^8 + 25*x^7 - 264*x^6 + 420*x^5 - 391*x^4 + 51*x^3 + 118*x^2 - 17*x - 11)
 
gp: K = bnfinit(x^20 - 3*x^17 - 23*x^16 + 46*x^15 - 62*x^14 + 306*x^13 - 395*x^12 + 394*x^11 - 779*x^10 + 893*x^9 - 548*x^8 + 25*x^7 - 264*x^6 + 420*x^5 - 391*x^4 + 51*x^3 + 118*x^2 - 17*x - 11, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{17} - 23 x^{16} + 46 x^{15} - 62 x^{14} + 306 x^{13} - 395 x^{12} + 394 x^{11} - 779 x^{10} + 893 x^{9} - 548 x^{8} + 25 x^{7} - 264 x^{6} + 420 x^{5} - 391 x^{4} + 51 x^{3} + 118 x^{2} - 17 x - 11 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(45215251470640684072265625=3^{10}\cdot 5^{10}\cdot 601^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.18$
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:  $3, 5, 601$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{7646928601422817915261765463333} a^{19} + \frac{1069797981128969277725676945981}{7646928601422817915261765463333} a^{18} + \frac{2623545851540916240684515310195}{7646928601422817915261765463333} a^{17} - \frac{1696022958550698203730684994796}{7646928601422817915261765463333} a^{16} + \frac{1772443882491297891949753698635}{7646928601422817915261765463333} a^{15} + \frac{2579268014169842441835623948771}{7646928601422817915261765463333} a^{14} - \frac{3051275939559169169109694214078}{7646928601422817915261765463333} a^{13} - \frac{1007805420121812256934944107264}{7646928601422817915261765463333} a^{12} + \frac{3622902958544945129944779811818}{7646928601422817915261765463333} a^{11} + \frac{1370115447494454051089403351694}{7646928601422817915261765463333} a^{10} - \frac{3407065334666573754116137977629}{7646928601422817915261765463333} a^{9} + \frac{1267537597449055384677583637343}{7646928601422817915261765463333} a^{8} - \frac{8990733541995824005256153215}{85920546083402448486087252397} a^{7} - \frac{591455012682388795458969859618}{7646928601422817915261765463333} a^{6} + \frac{3820675283489558221482968217228}{7646928601422817915261765463333} a^{5} - \frac{2830829753140804717183310808266}{7646928601422817915261765463333} a^{4} + \frac{10228110838762828008750103556}{85920546083402448486087252397} a^{3} - \frac{183020732026245147258116179063}{7646928601422817915261765463333} a^{2} + \frac{3286497779377203330904504546274}{7646928601422817915261765463333} a + \frac{1859576278430148146548326483423}{7646928601422817915261765463333}$

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:  $11$
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:  \( 72349.8991655 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_5\wr C_2$ (as 20T92):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $C_2\times D_5\wr C_2$
Character table for $C_2\times D_5\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.135225.1, 10.2.1128753125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
3Data not computed
5Data not computed
601Data not computed