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Properties

Label 20.20.6996929561...4544.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{24}\cdot 37^{10}\cdot 131^{8}$
Root discriminant $98.23$
Ramified primes $2, 37, 131$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T145

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, 164, -12081, 5686, 161555, -287006, -298358, 1108148, -619334, -674530, 905766, -195370, -187882, 98844, 3911, -11084, 1350, 392, -74, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 74*x^18 + 392*x^17 + 1350*x^16 - 11084*x^15 + 3911*x^14 + 98844*x^13 - 187882*x^12 - 195370*x^11 + 905766*x^10 - 674530*x^9 - 619334*x^8 + 1108148*x^7 - 298358*x^6 - 287006*x^5 + 161555*x^4 + 5686*x^3 - 12081*x^2 + 164*x + 7)
 
gp: K = bnfinit(x^20 - 4*x^19 - 74*x^18 + 392*x^17 + 1350*x^16 - 11084*x^15 + 3911*x^14 + 98844*x^13 - 187882*x^12 - 195370*x^11 + 905766*x^10 - 674530*x^9 - 619334*x^8 + 1108148*x^7 - 298358*x^6 - 287006*x^5 + 161555*x^4 + 5686*x^3 - 12081*x^2 + 164*x + 7, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 74 x^{18} + 392 x^{17} + 1350 x^{16} - 11084 x^{15} + 3911 x^{14} + 98844 x^{13} - 187882 x^{12} - 195370 x^{11} + 905766 x^{10} - 674530 x^{9} - 619334 x^{8} + 1108148 x^{7} - 298358 x^{6} - 287006 x^{5} + 161555 x^{4} + 5686 x^{3} - 12081 x^{2} + 164 x + 7 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6996929561265899684619283912699144044544=2^{24}\cdot 37^{10}\cdot 131^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.23$
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:  $2, 37, 131$
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}{7891703551053591815720312957043602827424483} a^{19} - \frac{2647333435032204166033605545039574624111075}{7891703551053591815720312957043602827424483} a^{18} + \frac{3625336723415162164220676807940602230662406}{7891703551053591815720312957043602827424483} a^{17} - \frac{3204411882845876886550581265999640539976399}{7891703551053591815720312957043602827424483} a^{16} + \frac{3061607600197556164122898959932211211520523}{7891703551053591815720312957043602827424483} a^{15} - \frac{673970289599218745838226514455223607886811}{7891703551053591815720312957043602827424483} a^{14} + \frac{3234994630597381634003398108282357490067754}{7891703551053591815720312957043602827424483} a^{13} - \frac{1128962665836430750144203734715309655037304}{7891703551053591815720312957043602827424483} a^{12} - \frac{2853988397921820244285213940752261464207098}{7891703551053591815720312957043602827424483} a^{11} - \frac{228933193355824044722668495451057495767086}{7891703551053591815720312957043602827424483} a^{10} - \frac{3669629161503627291412101841911001304708096}{7891703551053591815720312957043602827424483} a^{9} + \frac{1056081729418864486084987652649546407520760}{7891703551053591815720312957043602827424483} a^{8} + \frac{1174212695954584471709468066250739543255826}{7891703551053591815720312957043602827424483} a^{7} - \frac{2979249372696658242590115054869059651488476}{7891703551053591815720312957043602827424483} a^{6} + \frac{2628155365749574780411011536054735242213327}{7891703551053591815720312957043602827424483} a^{5} - \frac{917921346406827425963547761926318881985932}{7891703551053591815720312957043602827424483} a^{4} - \frac{1741726176227712251951376021751795770222404}{7891703551053591815720312957043602827424483} a^{3} - \frac{3702891528144727460074334078976237710144976}{7891703551053591815720312957043602827424483} a^{2} + \frac{2604550396459687709295055053859806769902410}{7891703551053591815720312957043602827424483} a + \frac{60268746771173476783822568357960013270037}{7891703551053591815720312957043602827424483}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  $19$
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:  \( 28197239596300 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T145:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 720
The 11 conjugacy class representatives for t20n145
Character table for t20n145

Intermediate fields

\(\Q(\sqrt{37}) \), 10.10.61101279225499648.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 6 siblings: 6.6.55632392512.1, 6.6.40637248.1
Degree 10 sibling: 10.10.61101279225499648.1
Degree 12 siblings: Deg 12, Deg 12
Degree 15 siblings: Deg 15, Deg 15
Degree 20 siblings: Deg 20, 20.20.3733366322992474864549675248123904.1
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.8.12.16$x^{8} + 24 x^{2} + 4$$4$$2$$12$$A_4\times C_2$$[2, 2]^{6}$
2.12.12.11$x^{12} - 6 x^{10} - 73 x^{8} + 140 x^{6} + 79 x^{4} - 6 x^{2} + 57$$2$$6$$12$$A_4 \times C_2$$[2, 2]^{6}$
37Data not computed
$131$131.2.0.1$x^{2} - x + 14$$1$$2$$0$$C_2$$[\ ]^{2}$
131.2.0.1$x^{2} - x + 14$$1$$2$$0$$C_2$$[\ ]^{2}$
131.4.2.1$x^{4} + 3537 x^{2} + 3363556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
131.4.2.1$x^{4} + 3537 x^{2} + 3363556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
131.4.2.1$x^{4} + 3537 x^{2} + 3363556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
131.4.2.1$x^{4} + 3537 x^{2} + 3363556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$