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Properties

Label 20.10.8925264962...0032.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{20}\cdot 3^{10}\cdot 7^{8}\cdot 11^{9}\cdot 13^{9}$
Root discriminant $70.39$
Ramified primes $2, 3, 7, 11, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{10}^2:C_2^2$ (as 20T106)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-78119, -996780, -3970514, -6818088, -4193927, 2146840, 3904306, 755896, -958240, -283164, 145308, 18456, -20145, 4360, 1262, -1168, 147, 116, -26, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 26*x^18 + 116*x^17 + 147*x^16 - 1168*x^15 + 1262*x^14 + 4360*x^13 - 20145*x^12 + 18456*x^11 + 145308*x^10 - 283164*x^9 - 958240*x^8 + 755896*x^7 + 3904306*x^6 + 2146840*x^5 - 4193927*x^4 - 6818088*x^3 - 3970514*x^2 - 996780*x - 78119)
 
gp: K = bnfinit(x^20 - 4*x^19 - 26*x^18 + 116*x^17 + 147*x^16 - 1168*x^15 + 1262*x^14 + 4360*x^13 - 20145*x^12 + 18456*x^11 + 145308*x^10 - 283164*x^9 - 958240*x^8 + 755896*x^7 + 3904306*x^6 + 2146840*x^5 - 4193927*x^4 - 6818088*x^3 - 3970514*x^2 - 996780*x - 78119, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 26 x^{18} + 116 x^{17} + 147 x^{16} - 1168 x^{15} + 1262 x^{14} + 4360 x^{13} - 20145 x^{12} + 18456 x^{11} + 145308 x^{10} - 283164 x^{9} - 958240 x^{8} + 755896 x^{7} + 3904306 x^{6} + 2146840 x^{5} - 4193927 x^{4} - 6818088 x^{3} - 3970514 x^{2} - 996780 x - 78119 \)

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:  \(-8925264962285150499914914463751340032=-\,2^{20}\cdot 3^{10}\cdot 7^{8}\cdot 11^{9}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.39$
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:  $2, 3, 7, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{14} + \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{15} + \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{52} a^{17} - \frac{1}{13} a^{16} - \frac{3}{26} a^{15} + \frac{1}{26} a^{14} - \frac{3}{26} a^{13} - \frac{1}{52} a^{12} - \frac{5}{52} a^{11} - \frac{3}{26} a^{10} - \frac{9}{52} a^{9} + \frac{5}{13} a^{8} - \frac{2}{13} a^{7} + \frac{19}{52} a^{6} - \frac{5}{52} a^{5} - \frac{1}{4} a^{4} + \frac{6}{13} a^{3} - \frac{9}{26} a^{2} + \frac{1}{26} a + \frac{3}{52}$, $\frac{1}{285844} a^{18} - \frac{55}{142922} a^{17} - \frac{20811}{285844} a^{16} - \frac{1953}{71461} a^{15} - \frac{967}{142922} a^{14} + \frac{16105}{285844} a^{13} + \frac{13031}{142922} a^{12} - \frac{2027}{71461} a^{11} - \frac{7303}{285844} a^{10} - \frac{19081}{71461} a^{9} - \frac{17093}{142922} a^{8} + \frac{69481}{285844} a^{7} + \frac{94779}{285844} a^{6} - \frac{47063}{285844} a^{5} + \frac{35340}{71461} a^{4} - \frac{10719}{142922} a^{3} - \frac{121733}{285844} a^{2} + \frac{97889}{285844} a + \frac{9651}{71461}$, $\frac{1}{180872778586471588080044240533809573450223996} a^{19} + \frac{120558571341621033257363576127755707883}{180872778586471588080044240533809573450223996} a^{18} - \frac{800451727113226684480793973134139238659621}{90436389293235794040022120266904786725111998} a^{17} + \frac{2569700971368111571550462143390031773997529}{180872778586471588080044240533809573450223996} a^{16} + \frac{382098813985080047742714933399759544811503}{45218194646617897020011060133452393362555999} a^{15} + \frac{20621056063273424600357323623467578958371163}{180872778586471588080044240533809573450223996} a^{14} - \frac{378393279664646832266612112201374155543278}{3478322665124453616923927702573261027888923} a^{13} + \frac{598590107659288054956826951452326267884601}{13913290660497814467695710810293044111555692} a^{12} - \frac{7363911396315207129373105319553434554680967}{180872778586471588080044240533809573450223996} a^{11} - \frac{19929801652357692546229252127727727828985777}{180872778586471588080044240533809573450223996} a^{10} - \frac{4882544141912006054257236140222147979006898}{45218194646617897020011060133452393362555999} a^{9} - \frac{40996287078001881346004118846034662379894399}{180872778586471588080044240533809573450223996} a^{8} + \frac{39345639896806218874441867005536090868188083}{90436389293235794040022120266904786725111998} a^{7} + \frac{29027590841465025226864766268940712593863997}{180872778586471588080044240533809573450223996} a^{6} - \frac{20423111608273872019926043073639703056076757}{180872778586471588080044240533809573450223996} a^{5} - \frac{69411850937168315520660721552049068874813121}{180872778586471588080044240533809573450223996} a^{4} - \frac{12640249262035347038885717230907888716718741}{45218194646617897020011060133452393362555999} a^{3} + \frac{7125319905459791380893829096052978100427584}{45218194646617897020011060133452393362555999} a^{2} - \frac{78855960320032705805519980965548647554052089}{180872778586471588080044240533809573450223996} a + \frac{73245830544849390464837146172570429653154987}{180872778586471588080044240533809573450223996}$

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:  \( 145012875501 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{10}^2:C_2^2$ (as 20T106):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 46 conjugacy class representatives for $C_{10}^2:C_2^2$
Character table for $C_{10}^2:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.2.20592.2, 10.10.249828821987576832.1

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 $20$ R R R $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ $20$ $20$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11.10.9.6$x^{10} + 216513$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$