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Properties

Label 20.8.91039466525...8681.1
Degree $20$
Signature $[8, 6]$
Discriminant $3^{2}\cdot 97^{2}\cdot 401^{10}$
Root discriminant $35.32$
Ramified primes $3, 97, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T347

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![397, 10407, 37499, -43180, -19797, 48253, -40646, 16124, 8514, -15811, 12573, -6527, 1705, 124, -446, 297, -89, 12, 3, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 3*x^18 + 12*x^17 - 89*x^16 + 297*x^15 - 446*x^14 + 124*x^13 + 1705*x^12 - 6527*x^11 + 12573*x^10 - 15811*x^9 + 8514*x^8 + 16124*x^7 - 40646*x^6 + 48253*x^5 - 19797*x^4 - 43180*x^3 + 37499*x^2 + 10407*x + 397)
 
gp: K = bnfinit(x^20 - 4*x^19 + 3*x^18 + 12*x^17 - 89*x^16 + 297*x^15 - 446*x^14 + 124*x^13 + 1705*x^12 - 6527*x^11 + 12573*x^10 - 15811*x^9 + 8514*x^8 + 16124*x^7 - 40646*x^6 + 48253*x^5 - 19797*x^4 - 43180*x^3 + 37499*x^2 + 10407*x + 397, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 3 x^{18} + 12 x^{17} - 89 x^{16} + 297 x^{15} - 446 x^{14} + 124 x^{13} + 1705 x^{12} - 6527 x^{11} + 12573 x^{10} - 15811 x^{9} + 8514 x^{8} + 16124 x^{7} - 40646 x^{6} + 48253 x^{5} - 19797 x^{4} - 43180 x^{3} + 37499 x^{2} + 10407 x + 397 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9103946652567430482896122008681=3^{2}\cdot 97^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.32$
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:  $3, 97, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{291} a^{18} - \frac{4}{97} a^{17} + \frac{36}{97} a^{16} + \frac{107}{291} a^{15} - \frac{70}{291} a^{14} - \frac{23}{291} a^{13} - \frac{19}{291} a^{12} - \frac{125}{291} a^{11} - \frac{85}{291} a^{10} + \frac{109}{291} a^{9} - \frac{122}{291} a^{8} - \frac{80}{291} a^{7} - \frac{92}{291} a^{6} + \frac{38}{291} a^{5} - \frac{68}{291} a^{4} - \frac{40}{291} a^{3} + \frac{29}{97} a^{2} - \frac{4}{291} a - \frac{1}{291}$, $\frac{1}{90292688044632754536051436862126256693} a^{19} - \frac{110287294176284424133886038539153329}{90292688044632754536051436862126256693} a^{18} + \frac{1842906300250115085931263081198099200}{30097562681544251512017145620708752231} a^{17} + \frac{43832937379193323726184368289344616462}{90292688044632754536051436862126256693} a^{16} - \frac{40214640815234568166957725357741897854}{90292688044632754536051436862126256693} a^{15} + \frac{14891710142825529028298963284370314456}{30097562681544251512017145620708752231} a^{14} - \frac{2059000459031685895230864475288980099}{30097562681544251512017145620708752231} a^{13} - \frac{5950260615299489560221545187847165471}{30097562681544251512017145620708752231} a^{12} + \frac{61179660569153327368555519617145561}{129174088761992495759730238715488207} a^{11} - \frac{12403473556045100883830279594607549037}{30097562681544251512017145620708752231} a^{10} - \frac{856056328421754417532507120850226446}{5311334590860750266826555109536838629} a^{9} - \frac{36444888558676230127523666561840672476}{90292688044632754536051436862126256693} a^{8} - \frac{1413467240281967405598030486071204684}{5311334590860750266826555109536838629} a^{7} + \frac{1892911846581968865433606233706925997}{30097562681544251512017145620708752231} a^{6} + \frac{10638237029640374222489236907449487654}{30097562681544251512017145620708752231} a^{5} - \frac{9289749897109188968230047129737066481}{30097562681544251512017145620708752231} a^{4} - \frac{1928222216052070855795290295439789348}{3925769045418815414610932037483750291} a^{3} + \frac{24979782974131573758978901395016717070}{90292688044632754536051436862126256693} a^{2} + \frac{5954846681901762638685337741598171851}{90292688044632754536051436862126256693} a + \frac{18601073136518884623952621553509867709}{90292688044632754536051436862126256693}$

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:  $13$
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:  \( 91455948.851 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T347:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n347 are not computed
Character table for t20n347 is not computed

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed