↑

Properties

Label 20.20.3575798800...6544.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{10}\cdot 3^{4}\cdot 401^{11}$
Root discriminant $47.61$
Ramified primes $2, 3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T350

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9829, -11610, 85962, 140766, -177957, -418522, 54268, 478773, 125534, -249104, -114182, 66143, 40876, -9340, -7675, 668, 800, -19, -44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 44*x^18 - 19*x^17 + 800*x^16 + 668*x^15 - 7675*x^14 - 9340*x^13 + 40876*x^12 + 66143*x^11 - 114182*x^10 - 249104*x^9 + 125534*x^8 + 478773*x^7 + 54268*x^6 - 418522*x^5 - 177957*x^4 + 140766*x^3 + 85962*x^2 - 11610*x - 9829)
 
gp: K = bnfinit(x^20 - 44*x^18 - 19*x^17 + 800*x^16 + 668*x^15 - 7675*x^14 - 9340*x^13 + 40876*x^12 + 66143*x^11 - 114182*x^10 - 249104*x^9 + 125534*x^8 + 478773*x^7 + 54268*x^6 - 418522*x^5 - 177957*x^4 + 140766*x^3 + 85962*x^2 - 11610*x - 9829, 1)
 

Normalized defining polynomial

\( x^{20} - 44 x^{18} - 19 x^{17} + 800 x^{16} + 668 x^{15} - 7675 x^{14} - 9340 x^{13} + 40876 x^{12} + 66143 x^{11} - 114182 x^{10} - 249104 x^{9} + 125534 x^{8} + 478773 x^{7} + 54268 x^{6} - 418522 x^{5} - 177957 x^{4} + 140766 x^{3} + 85962 x^{2} - 11610 x - 9829 \)

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:  \(3575798800337404311986250912236544=2^{10}\cdot 3^{4}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3081} a^{18} + \frac{194}{3081} a^{17} - \frac{328}{1027} a^{16} + \frac{112}{3081} a^{15} - \frac{1256}{3081} a^{14} - \frac{547}{3081} a^{13} - \frac{146}{3081} a^{12} - \frac{463}{1027} a^{11} - \frac{568}{3081} a^{10} - \frac{174}{1027} a^{9} - \frac{228}{1027} a^{8} - \frac{500}{3081} a^{7} + \frac{85}{237} a^{6} + \frac{22}{79} a^{5} + \frac{1124}{3081} a^{4} + \frac{258}{1027} a^{3} - \frac{583}{3081} a^{2} + \frac{82}{3081} a - \frac{1295}{3081}$, $\frac{1}{9704765939598003398763249} a^{19} + \frac{694001692605410200300}{9704765939598003398763249} a^{18} + \frac{70296579087249684752363}{746520456892154107597173} a^{17} - \frac{1612887090940030041346316}{9704765939598003398763249} a^{16} + \frac{1470014149854574531157270}{3234921979866001132921083} a^{15} - \frac{2369260015963259680904455}{9704765939598003398763249} a^{14} + \frac{4503917777355964605388120}{9704765939598003398763249} a^{13} + \frac{382891739822963297229148}{3234921979866001132921083} a^{12} + \frac{2803821658220391725312200}{9704765939598003398763249} a^{11} - \frac{85692313210044898762952}{190289528227411831348299} a^{10} + \frac{2839642838718220762254700}{9704765939598003398763249} a^{9} - \frac{1951610313301273070155540}{9704765939598003398763249} a^{8} + \frac{3341357564971561503876835}{9704765939598003398763249} a^{7} - \frac{4628261096550250228061}{9449626036609545665787} a^{6} + \frac{793383654708164701256744}{9704765939598003398763249} a^{5} + \frac{4376659632519833970203560}{9704765939598003398763249} a^{4} + \frac{2342597882001580914724186}{9704765939598003398763249} a^{3} + \frac{217214283358389825078626}{570868584682235494044897} a^{2} - \frac{2219863864203670771890419}{9704765939598003398763249} a + \frac{1877591907529737801241561}{9704765939598003398763249}$

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:  \( 24698442529.8 \) (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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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)$$[2, 2, 2, 2]^{10}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{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}$
3.4.2.1$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}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed