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Properties

Label 20.4.65229783036...0000.2
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 5^{14}\cdot 269^{7}$
Root discriminant $43.72$
Ramified primes $2, 5, 269$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T375

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19465109, 0, 19537470, 0, 2578365, 0, -1623110, 0, 116830, 0, -99921, 0, -6895, 0, 335, 0, -125, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 125*x^16 + 335*x^14 - 6895*x^12 - 99921*x^10 + 116830*x^8 - 1623110*x^6 + 2578365*x^4 + 19537470*x^2 + 19465109)
 
gp: K = bnfinit(x^20 - 125*x^16 + 335*x^14 - 6895*x^12 - 99921*x^10 + 116830*x^8 - 1623110*x^6 + 2578365*x^4 + 19537470*x^2 + 19465109, 1)
 

Normalized defining polynomial

\( x^{20} - 125 x^{16} + 335 x^{14} - 6895 x^{12} - 99921 x^{10} + 116830 x^{8} - 1623110 x^{6} + 2578365 x^{4} + 19537470 x^{2} + 19465109 \)

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:  \(652297830366246329600000000000000=2^{20}\cdot 5^{14}\cdot 269^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.72$
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:  $2, 5, 269$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{10} + \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{2}{5} a^{3} - \frac{1}{5} a$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{10} + \frac{2}{5} a^{4} + \frac{2}{5}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{11} + \frac{2}{5} a^{5} + \frac{2}{5} a$, $\frac{1}{20175} a^{16} - \frac{1}{75} a^{14} - \frac{221}{6725} a^{12} - \frac{1054}{6725} a^{10} + \frac{1849}{4035} a^{8} + \frac{2837}{20175} a^{6} - \frac{6103}{20175} a^{4} - \frac{2116}{20175} a^{2} + \frac{14}{75}$, $\frac{1}{20175} a^{17} - \frac{1}{75} a^{15} - \frac{221}{6725} a^{13} - \frac{1054}{6725} a^{11} + \frac{1849}{4035} a^{9} + \frac{2837}{20175} a^{7} - \frac{6103}{20175} a^{5} - \frac{2116}{20175} a^{3} + \frac{14}{75} a$, $\frac{1}{33445232204049042413083144998825} a^{18} + \frac{1733859939471544388633588}{124331718230665585178747750925} a^{16} - \frac{710680535684898582931799185134}{11148410734683014137694381666275} a^{14} - \frac{102011421724982997743535369709}{2229682146936602827538876333255} a^{12} + \frac{5143652132912945494751669211488}{33445232204049042413083144998825} a^{10} + \frac{830021721751862808343266553532}{33445232204049042413083144998825} a^{8} - \frac{2668253345281687193238762656986}{33445232204049042413083144998825} a^{6} - \frac{14861492309552223728480622451609}{33445232204049042413083144998825} a^{4} - \frac{4043263191392671748532727481}{24866343646133117035749550185} a^{2} + \frac{30760834488929383305454522}{154066565341593042352847275}$, $\frac{1}{33445232204049042413083144998825} a^{19} + \frac{1733859939471544388633588}{124331718230665585178747750925} a^{17} - \frac{710680535684898582931799185134}{11148410734683014137694381666275} a^{15} - \frac{102011421724982997743535369709}{2229682146936602827538876333255} a^{13} + \frac{5143652132912945494751669211488}{33445232204049042413083144998825} a^{11} + \frac{830021721751862808343266553532}{33445232204049042413083144998825} a^{9} - \frac{2668253345281687193238762656986}{33445232204049042413083144998825} a^{7} - \frac{14861492309552223728480622451609}{33445232204049042413083144998825} a^{5} - \frac{4043263191392671748532727481}{24866343646133117035749550185} a^{3} + \frac{30760834488929383305454522}{154066565341593042352847275} a$

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:  \( 227708300.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T375:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 48 conjugacy class representatives for t20n375
Character table for t20n375 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.33625.1, 10.2.5653203125.1

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
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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$5.4.2.1$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}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
269Data not computed