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Properties

Label 20.10.1130629936...0496.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{34}\cdot 7^{10}\cdot 13^{12}$
Root discriminant $40.06$
Ramified primes $2, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T633

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4151, -5298, -26143, 6644, 54896, -1574, -60698, 3656, 41092, -7992, -17635, 6108, 4427, -2172, -425, 306, -45, 16, 5, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 5*x^18 + 16*x^17 - 45*x^16 + 306*x^15 - 425*x^14 - 2172*x^13 + 4427*x^12 + 6108*x^11 - 17635*x^10 - 7992*x^9 + 41092*x^8 + 3656*x^7 - 60698*x^6 - 1574*x^5 + 54896*x^4 + 6644*x^3 - 26143*x^2 - 5298*x + 4151)
 
gp: K = bnfinit(x^20 - 6*x^19 + 5*x^18 + 16*x^17 - 45*x^16 + 306*x^15 - 425*x^14 - 2172*x^13 + 4427*x^12 + 6108*x^11 - 17635*x^10 - 7992*x^9 + 41092*x^8 + 3656*x^7 - 60698*x^6 - 1574*x^5 + 54896*x^4 + 6644*x^3 - 26143*x^2 - 5298*x + 4151, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 5 x^{18} + 16 x^{17} - 45 x^{16} + 306 x^{15} - 425 x^{14} - 2172 x^{13} + 4427 x^{12} + 6108 x^{11} - 17635 x^{10} - 7992 x^{9} + 41092 x^{8} + 3656 x^{7} - 60698 x^{6} - 1574 x^{5} + 54896 x^{4} + 6644 x^{3} - 26143 x^{2} - 5298 x + 4151 \)

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:  \(-113062993649232013634145926250496=-\,2^{34}\cdot 7^{10}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.06$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} + \frac{1}{23} a^{17} + \frac{11}{23} a^{16} - \frac{10}{23} a^{14} + \frac{6}{23} a^{13} - \frac{5}{23} a^{12} - \frac{5}{23} a^{11} + \frac{4}{23} a^{10} + \frac{2}{23} a^{8} + \frac{3}{23} a^{7} + \frac{10}{23} a^{6} - \frac{3}{23} a^{5} - \frac{9}{23} a^{4} - \frac{1}{23} a^{3} - \frac{3}{23} a^{2} + \frac{11}{23}$, $\frac{1}{2313425165609588618507759250360387991} a^{19} - \frac{46733588774888802092487927744643454}{2313425165609588618507759250360387991} a^{18} + \frac{41761782070516281518266468789696005}{330489309372798374072537035765769713} a^{17} + \frac{22124566674142571396230884819689157}{2313425165609588618507759250360387991} a^{16} - \frac{840978304365541322416297556896908400}{2313425165609588618507759250360387991} a^{15} + \frac{889832064010696169984798624597964440}{2313425165609588618507759250360387991} a^{14} + \frac{203966445577039570785390607472991535}{2313425165609588618507759250360387991} a^{13} - \frac{438858794753011662862328011297434194}{2313425165609588618507759250360387991} a^{12} + \frac{120418025819438256853319744272908156}{330489309372798374072537035765769713} a^{11} + \frac{457261822918752762629248657469606996}{2313425165609588618507759250360387991} a^{10} - \frac{283292567969763904647366508216510872}{2313425165609588618507759250360387991} a^{9} - \frac{417348596025478640634485090224911609}{2313425165609588618507759250360387991} a^{8} + \frac{1063621212424083538746520190844011505}{2313425165609588618507759250360387991} a^{7} + \frac{1126898072682591859765445679679371805}{2313425165609588618507759250360387991} a^{6} - \frac{101368227070078421868087938456477062}{330489309372798374072537035765769713} a^{5} + \frac{680523896242863527908439628806320520}{2313425165609588618507759250360387991} a^{4} - \frac{952556993630608222739208576389527959}{2313425165609588618507759250360387991} a^{3} + \frac{1128431942416539593355326068950608529}{2313425165609588618507759250360387991} a^{2} - \frac{134232063756252327950431112460232392}{2313425165609588618507759250360387991} a + \frac{163657130854655805491865622032164011}{330489309372798374072537035765769713}$

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:  \( 1171729471.05 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T633:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40960
The 124 conjugacy class representatives for t20n633 are not computed
Character table for t20n633 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.5.6889792.1, 10.10.379753870426112.1

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 $20$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.6.3$x^{8} - 7 x^{4} + 147$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
13Data not computed