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Properties

Label 20.12.2549038515...0000.8
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{15}\cdot 6029^{5}$
Root discriminant $58.93$
Ramified primes $2, 5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![753625, 0, 1507250, 0, -377050, 0, -878600, 0, 397185, 0, -6435, 0, -21014, 0, 3064, 0, 31, 0, -26, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 26*x^18 + 31*x^16 + 3064*x^14 - 21014*x^12 - 6435*x^10 + 397185*x^8 - 878600*x^6 - 377050*x^4 + 1507250*x^2 + 753625)
 
gp: K = bnfinit(x^20 - 26*x^18 + 31*x^16 + 3064*x^14 - 21014*x^12 - 6435*x^10 + 397185*x^8 - 878600*x^6 - 377050*x^4 + 1507250*x^2 + 753625, 1)
 

Normalized defining polynomial

\( x^{20} - 26 x^{18} + 31 x^{16} + 3064 x^{14} - 21014 x^{12} - 6435 x^{10} + 397185 x^{8} - 878600 x^{6} - 377050 x^{4} + 1507250 x^{2} + 753625 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(254903851560926116768000000000000000=2^{20}\cdot 5^{15}\cdot 6029^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.93$
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:  $2, 5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{6} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{7} + \frac{1}{5} a^{5}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{6} - \frac{2}{5} a^{4}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{7} - \frac{2}{5} a^{5}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{6} - \frac{2}{5} a^{4}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{7} - \frac{2}{5} a^{5}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{5}$, $\frac{1}{2275} a^{16} + \frac{204}{2275} a^{14} - \frac{79}{2275} a^{12} - \frac{81}{2275} a^{10} - \frac{219}{2275} a^{8} - \frac{45}{91} a^{6} - \frac{121}{455} a^{4} + \frac{3}{13} a^{2} + \frac{20}{91}$, $\frac{1}{2275} a^{17} + \frac{204}{2275} a^{15} - \frac{79}{2275} a^{13} - \frac{81}{2275} a^{11} - \frac{219}{2275} a^{9} - \frac{45}{91} a^{7} - \frac{121}{455} a^{5} + \frac{3}{13} a^{3} + \frac{20}{91} a$, $\frac{1}{35785593353586124327825} a^{18} + \frac{1031746965968811699}{5112227621940874903975} a^{16} - \frac{164531837294135431298}{35785593353586124327825} a^{14} + \frac{272659432037772965323}{35785593353586124327825} a^{12} + \frac{1112859250058726781802}{35785593353586124327825} a^{10} - \frac{3573067024476503333146}{35785593353586124327825} a^{8} + \frac{2408143998780965180672}{7157118670717224865565} a^{6} - \frac{931288900471617262426}{7157118670717224865565} a^{4} + \frac{560753158516569872188}{1431423734143444973113} a^{2} + \frac{674512250288111893386}{1431423734143444973113}$, $\frac{1}{178927966767930621639125} a^{19} + \frac{38682091050648604379}{178927966767930621639125} a^{17} - \frac{903838601082508109719}{178927966767930621639125} a^{15} - \frac{9369788359499938776636}{178927966767930621639125} a^{13} - \frac{1435389595339493939564}{178927966767930621639125} a^{11} + \frac{314530832754666149689}{5112227621940874903975} a^{9} + \frac{12506759793507247298678}{35785593353586124327825} a^{7} + \frac{612080001623348589614}{1431423734143444973113} a^{5} - \frac{210013467560669728719}{7157118670717224865565} a^{3} - \frac{311827594444287920624}{1431423734143444973113} a$

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:  $15$
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:  \( 34145971559.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n797 are not computed
Character table for t20n797 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
6029Data not computed