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Properties

Label 20.0.98980918726...0256.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{2}\cdot 97^{2}\cdot 401^{15}$
Root discriminant $316.07$
Ramified primes $2, 3, 97, 401$
Class number $66058729472$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 4, 258041912]$ (GRH)
Galois group 20T350

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![878026939499046681, 0, 11444522960549429763, 0, 1076119346023005367, 0, 41765095228552836, 0, 870171034803357, 0, 10700977711955, 0, 80925997668, 0, 378364753, 0, 1057838, 0, 1604, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 1604*x^18 + 1057838*x^16 + 378364753*x^14 + 80925997668*x^12 + 10700977711955*x^10 + 870171034803357*x^8 + 41765095228552836*x^6 + 1076119346023005367*x^4 + 11444522960549429763*x^2 + 878026939499046681)
 
gp: K = bnfinit(x^20 + 1604*x^18 + 1057838*x^16 + 378364753*x^14 + 80925997668*x^12 + 10700977711955*x^10 + 870171034803357*x^8 + 41765095228552836*x^6 + 1076119346023005367*x^4 + 11444522960549429763*x^2 + 878026939499046681, 1)
 

Normalized defining polynomial

\( x^{20} + 1604 x^{18} + 1057838 x^{16} + 378364753 x^{14} + 80925997668 x^{12} + 10700977711955 x^{10} + 870171034803357 x^{8} + 41765095228552836 x^{6} + 1076119346023005367 x^{4} + 11444522960549429763 x^{2} + 878026939499046681 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(98980918726972831182987982310379899184973800800256=2^{20}\cdot 3^{2}\cdot 97^{2}\cdot 401^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $316.07$
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:  $2, 3, 97, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{401} a^{4}$, $\frac{1}{401} a^{5}$, $\frac{1}{401} a^{6}$, $\frac{1}{401} a^{7}$, $\frac{1}{160801} a^{8}$, $\frac{1}{160801} a^{9}$, $\frac{1}{160801} a^{10}$, $\frac{1}{160801} a^{11}$, $\frac{1}{193443603} a^{12} - \frac{1}{482403} a^{10} - \frac{1}{482403} a^{8} - \frac{1}{1203} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{193443603} a^{13} - \frac{1}{482403} a^{11} - \frac{1}{482403} a^{9} - \frac{1}{1203} a^{7} - \frac{1}{3} a^{3}$, $\frac{1}{193443603} a^{14} - \frac{1}{482403} a^{8} + \frac{1}{1203} a^{6} + \frac{1}{1203} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{193443603} a^{15} - \frac{1}{482403} a^{9} + \frac{1}{1203} a^{7} + \frac{1}{1203} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{77570884803} a^{16} + \frac{1}{482403} a^{10} + \frac{1}{482403} a^{8} - \frac{1}{1203} a^{6} + \frac{1}{1203} a^{4}$, $\frac{1}{77570884803} a^{17} + \frac{1}{482403} a^{11} + \frac{1}{482403} a^{9} - \frac{1}{1203} a^{7} + \frac{1}{1203} a^{5}$, $\frac{1}{6167002734873578673417030341956594119387} a^{18} - \frac{9227793806935593675817090013}{2055667578291192891139010113985531373129} a^{16} + \frac{56239290170381740837044122}{69588503118601445181356905721630247} a^{14} + \frac{979689039886084862744761759}{394334851005408189361022465755904733} a^{12} + \frac{3840261187844507455881414866}{12783922850549392672551850510789929} a^{10} - \frac{49535652012409970203741592125}{38351768551648178017655551532369787} a^{8} + \frac{118865056291309886500604910778}{95640320577676254408118582374987} a^{6} + \frac{43441979140687274189482952635}{95640320577676254408118582374987} a^{4} - \frac{23285742882553307528730988553}{79501513364651915551220766729} a^{2} - \frac{37585457125056866721274185}{273201076854473936602133219}$, $\frac{1}{6167002734873578673417030341956594119387} a^{19} - \frac{9227793806935593675817090013}{2055667578291192891139010113985531373129} a^{17} + \frac{56239290170381740837044122}{69588503118601445181356905721630247} a^{15} + \frac{979689039886084862744761759}{394334851005408189361022465755904733} a^{13} + \frac{3840261187844507455881414866}{12783922850549392672551850510789929} a^{11} - \frac{49535652012409970203741592125}{38351768551648178017655551532369787} a^{9} + \frac{118865056291309886500604910778}{95640320577676254408118582374987} a^{7} + \frac{43441979140687274189482952635}{95640320577676254408118582374987} a^{5} - \frac{23285742882553307528730988553}{79501513364651915551220766729} a^{3} - \frac{37585457125056866721274185}{273201076854473936602133219} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{258041912}$, which has order $66058729472$ (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:  $9$
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:  \( 795087.603907 \) (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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.10.10.1$x^{10} - 9 x^{8} + 54 x^{6} - 38 x^{4} + 41 x^{2} - 17$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
2.10.10.2$x^{10} - 5 x^{8} + 10 x^{6} - 2 x^{4} - 11 x^{2} + 39$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
$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}$
3.4.2.2$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}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
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}$
401Data not computed