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Properties

Label 20.4.25340825466...4833.1
Degree $20$
Signature $[4, 8]$
Discriminant $17^{15}\cdot 97^{4}$
Root discriminant $20.90$
Ramified primes $17, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\times S_5$ (as 20T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-169, 403, 443, -1531, -1038, 7141, -11903, 12663, -10176, 6829, -4464, 2613, -1129, 436, -132, 6, -5, -5, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 5*x^18 - 5*x^17 - 5*x^16 + 6*x^15 - 132*x^14 + 436*x^13 - 1129*x^12 + 2613*x^11 - 4464*x^10 + 6829*x^9 - 10176*x^8 + 12663*x^7 - 11903*x^6 + 7141*x^5 - 1038*x^4 - 1531*x^3 + 443*x^2 + 403*x - 169)
 
gp: K = bnfinit(x^20 - x^19 + 5*x^18 - 5*x^17 - 5*x^16 + 6*x^15 - 132*x^14 + 436*x^13 - 1129*x^12 + 2613*x^11 - 4464*x^10 + 6829*x^9 - 10176*x^8 + 12663*x^7 - 11903*x^6 + 7141*x^5 - 1038*x^4 - 1531*x^3 + 443*x^2 + 403*x - 169, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 5 x^{18} - 5 x^{17} - 5 x^{16} + 6 x^{15} - 132 x^{14} + 436 x^{13} - 1129 x^{12} + 2613 x^{11} - 4464 x^{10} + 6829 x^{9} - 10176 x^{8} + 12663 x^{7} - 11903 x^{6} + 7141 x^{5} - 1038 x^{4} - 1531 x^{3} + 443 x^{2} + 403 x - 169 \)

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:  \(253408254667989956596734833=17^{15}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.90$
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:  $17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3947503104103853912771907700119017} a^{19} + \frac{961719605957558092344003438795127}{3947503104103853912771907700119017} a^{18} + \frac{964521652304387496886003184497313}{3947503104103853912771907700119017} a^{17} + \frac{1563714015716790655895544703410966}{3947503104103853912771907700119017} a^{16} + \frac{781216188335501483631018059065200}{3947503104103853912771907700119017} a^{15} - \frac{292934439735626245935624562111257}{3947503104103853912771907700119017} a^{14} + \frac{1430434250047104035817299829431882}{3947503104103853912771907700119017} a^{13} + \frac{1481776140036566965931475042328090}{3947503104103853912771907700119017} a^{12} + \frac{1489689584479051625946142540117732}{3947503104103853912771907700119017} a^{11} + \frac{42778449339068357288248812204678}{303654084931065685597839053855309} a^{10} - \frac{504753400733481019870002157622320}{3947503104103853912771907700119017} a^{9} - \frac{611015076849253789612802783006954}{3947503104103853912771907700119017} a^{8} + \frac{332470876165164178332359758938132}{3947503104103853912771907700119017} a^{7} - \frac{1094775971697531339529044991123162}{3947503104103853912771907700119017} a^{6} - \frac{992142522617131234679896843096009}{3947503104103853912771907700119017} a^{5} - \frac{1530869665268133187756396592681554}{3947503104103853912771907700119017} a^{4} + \frac{1805105050460787666027681432612582}{3947503104103853912771907700119017} a^{3} - \frac{261800980262029820870601922555675}{3947503104103853912771907700119017} a^{2} - \frac{143891581763047031532362967498222}{3947503104103853912771907700119017} a - \frac{51183596629136194999757964036643}{303654084931065685597839053855309}$

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:  \( 152775.352277 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times S_5$ (as 20T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 480
The 28 conjugacy class representatives for $C_4\times S_5$
Character table for $C_4\times S_5$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 5.1.1649.1, 10.2.13359434513.1

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 24 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ $20$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ $20$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.12.9.1$x^{12} - 34 x^{8} - 10115 x^{4} - 397953$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$