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Properties

Label 20.10.4305926880...7883.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,3^{5}\cdot 61^{10}\cdot 397^{4}$
Root discriminant $34.02$
Ramified primes $3, 61, 397$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T466

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-61, -122, 354, 97, 279, -2398, 846, 7423, -11868, 3223, 8524, -9305, 2079, 2175, -1529, 296, -55, 54, -10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 10*x^18 + 54*x^17 - 55*x^16 + 296*x^15 - 1529*x^14 + 2175*x^13 + 2079*x^12 - 9305*x^11 + 8524*x^10 + 3223*x^9 - 11868*x^8 + 7423*x^7 + 846*x^6 - 2398*x^5 + 279*x^4 + 97*x^3 + 354*x^2 - 122*x - 61)
 
gp: K = bnfinit(x^20 - 4*x^19 - 10*x^18 + 54*x^17 - 55*x^16 + 296*x^15 - 1529*x^14 + 2175*x^13 + 2079*x^12 - 9305*x^11 + 8524*x^10 + 3223*x^9 - 11868*x^8 + 7423*x^7 + 846*x^6 - 2398*x^5 + 279*x^4 + 97*x^3 + 354*x^2 - 122*x - 61, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 10 x^{18} + 54 x^{17} - 55 x^{16} + 296 x^{15} - 1529 x^{14} + 2175 x^{13} + 2079 x^{12} - 9305 x^{11} + 8524 x^{10} + 3223 x^{9} - 11868 x^{8} + 7423 x^{7} + 846 x^{6} - 2398 x^{5} + 279 x^{4} + 97 x^{3} + 354 x^{2} - 122 x - 61 \)

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:  \(-4305926880688359795034353497883=-\,3^{5}\cdot 61^{10}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.02$
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:  $3, 61, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11} a^{18} - \frac{5}{11} a^{17} - \frac{2}{11} a^{16} - \frac{3}{11} a^{15} - \frac{3}{11} a^{14} + \frac{4}{11} a^{13} - \frac{2}{11} a^{12} + \frac{5}{11} a^{10} - \frac{4}{11} a^{9} - \frac{4}{11} a^{8} + \frac{3}{11} a^{7} - \frac{3}{11} a^{6} - \frac{1}{11} a^{5} + \frac{2}{11} a^{4} - \frac{5}{11} a^{3} + \frac{4}{11} a^{2} + \frac{1}{11} a + \frac{2}{11}$, $\frac{1}{544287981329147463021484070519} a^{19} + \frac{737304851250838031664430458}{49480725575377042092862188229} a^{18} + \frac{29407518640791826193941163679}{544287981329147463021484070519} a^{17} + \frac{73270418085468234611888060316}{544287981329147463021484070519} a^{16} + \frac{133507735636266039667547420320}{544287981329147463021484070519} a^{15} - \frac{6691303439846546526970980380}{49480725575377042092862188229} a^{14} - \frac{133918855051744636993463180313}{544287981329147463021484070519} a^{13} - \frac{682597325780524335436870707}{3146173302480621173534590003} a^{12} + \frac{242388919749193974443377986982}{544287981329147463021484070519} a^{11} - \frac{84207677156859820947664725269}{544287981329147463021484070519} a^{10} + \frac{192718950673244365303070634858}{544287981329147463021484070519} a^{9} + \frac{246643100675233886634199605146}{544287981329147463021484070519} a^{8} + \frac{62936923735441738127137631772}{544287981329147463021484070519} a^{7} - \frac{201486933111608265039470867639}{544287981329147463021484070519} a^{6} - \frac{67039426572437197469238951889}{544287981329147463021484070519} a^{5} - \frac{41872333579452853417302332812}{544287981329147463021484070519} a^{4} - \frac{51875769089078808599546068522}{544287981329147463021484070519} a^{3} + \frac{13015813281363919939780098681}{544287981329147463021484070519} a^{2} + \frac{196502754514424533158122958452}{544287981329147463021484070519} a - \frac{219569706928521090947748433536}{544287981329147463021484070519}$

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:  \( 111873681.334 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 is not computed

Intermediate fields

\(\Q(\sqrt{61}) \), 5.5.24217.1, 10.10.133115978404309.1

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 $20$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$3$3.5.0.1$x^{5} - x + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
3.5.0.1$x^{5} - x + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
61Data not computed
397Data not computed