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Properties

Label 20.20.3661617971...1056.2
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 3^{4}\cdot 401^{11}$
Root discriminant $67.33$
Ramified primes $2, 3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T350

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32481, 0, -498042, 0, 1910421, 0, -2984028, 0, 2178056, 0, -850982, 0, 190169, 0, -24682, 0, 1807, 0, -68, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 68*x^18 + 1807*x^16 - 24682*x^14 + 190169*x^12 - 850982*x^10 + 2178056*x^8 - 2984028*x^6 + 1910421*x^4 - 498042*x^2 + 32481)
 
gp: K = bnfinit(x^20 - 68*x^18 + 1807*x^16 - 24682*x^14 + 190169*x^12 - 850982*x^10 + 2178056*x^8 - 2984028*x^6 + 1910421*x^4 - 498042*x^2 + 32481, 1)
 

Normalized defining polynomial

\( x^{20} - 68 x^{18} + 1807 x^{16} - 24682 x^{14} + 190169 x^{12} - 850982 x^{10} + 2178056 x^{8} - 2984028 x^{6} + 1910421 x^{4} - 498042 x^{2} + 32481 \)

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:  \(3661617971545502015473920934130221056=2^{20}\cdot 3^{4}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.33$
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:  $2, 3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{8} - \frac{1}{2} a^{6} - \frac{1}{3} a^{4} - \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{3} a^{5} - \frac{1}{6} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{12} + \frac{1}{3} a^{8} + \frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{13} + \frac{1}{3} a^{9} + \frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{8} + \frac{1}{6} a^{6} + \frac{1}{3} a^{4} - \frac{1}{6} a^{2}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{9} + \frac{1}{6} a^{7} + \frac{1}{3} a^{5} - \frac{1}{6} a^{3}$, $\frac{1}{46314} a^{16} + \frac{1399}{46314} a^{14} - \frac{2567}{46314} a^{12} + \frac{905}{46314} a^{10} + \frac{7519}{23157} a^{8} + \frac{7310}{23157} a^{6} + \frac{6337}{23157} a^{4} + \frac{2677}{15438} a^{2} - \frac{2373}{5146}$, $\frac{1}{46314} a^{17} + \frac{1399}{46314} a^{15} - \frac{2567}{46314} a^{13} + \frac{905}{46314} a^{11} + \frac{7519}{23157} a^{9} + \frac{7310}{23157} a^{7} + \frac{6337}{23157} a^{5} + \frac{2677}{15438} a^{3} - \frac{2373}{5146} a$, $\frac{1}{280347158478558366} a^{18} + \frac{753626727140}{140173579239279183} a^{16} + \frac{11198103222932111}{140173579239279183} a^{14} - \frac{23602532639912}{140173579239279183} a^{12} + \frac{11102988492566461}{140173579239279183} a^{10} - \frac{15932403633809045}{280347158478558366} a^{8} + \frac{21183199571942297}{280347158478558366} a^{6} + \frac{9001957600846733}{93449052826186122} a^{4} + \frac{2201743141865693}{31149684275395374} a^{2} - \frac{2014875191026466}{5191614045899229}$, $\frac{1}{280347158478558366} a^{19} + \frac{753626727140}{140173579239279183} a^{17} + \frac{11198103222932111}{140173579239279183} a^{15} - \frac{23602532639912}{140173579239279183} a^{13} + \frac{11102988492566461}{140173579239279183} a^{11} - \frac{15932403633809045}{280347158478558366} a^{9} + \frac{21183199571942297}{280347158478558366} a^{7} + \frac{9001957600846733}{93449052826186122} a^{5} + \frac{2201743141865693}{31149684275395374} a^{3} - \frac{2014875191026466}{5191614045899229} a$

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:  $19$
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:  \( 1022023448540 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T350:

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 t20n350 are not computed
Character table for t20n350 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 R R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.10.10.4$x^{10} - 5 x^{8} + 14 x^{6} - 22 x^{4} + 17 x^{2} - 37$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
$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$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed