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Properties

Label 20.10.2969955513...0000.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{20}\cdot 3^{2}\cdot 5^{10}\cdot 11\cdot 19^{8}\cdot 29^{7}$
Root discriminant $59.38$
Ramified primes $2, 3, 5, 11, 19, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T955

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-45936, 0, -180768, 0, 465539, 0, 69764, 0, -78088, 0, -16737, 0, 4482, 0, -347, 0, 133, 0, -22, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 22*x^18 + 133*x^16 - 347*x^14 + 4482*x^12 - 16737*x^10 - 78088*x^8 + 69764*x^6 + 465539*x^4 - 180768*x^2 - 45936)
 
gp: K = bnfinit(x^20 - 22*x^18 + 133*x^16 - 347*x^14 + 4482*x^12 - 16737*x^10 - 78088*x^8 + 69764*x^6 + 465539*x^4 - 180768*x^2 - 45936, 1)
 

Normalized defining polynomial

\( x^{20} - 22 x^{18} + 133 x^{16} - 347 x^{14} + 4482 x^{12} - 16737 x^{10} - 78088 x^{8} + 69764 x^{6} + 465539 x^{4} - 180768 x^{2} - 45936 \)

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:  \(-296995551360942445273405440000000000=-\,2^{20}\cdot 3^{2}\cdot 5^{10}\cdot 11\cdot 19^{8}\cdot 29^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.38$
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:  $2, 3, 5, 11, 19, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{3450709015549023926571003864} a^{18} - \frac{649811127053677093243197053}{1725354507774511963285501932} a^{16} + \frac{353830078166410224512643829}{3450709015549023926571003864} a^{14} - \frac{578639135144714617010389583}{3450709015549023926571003864} a^{12} - \frac{282884680093681512375053659}{575118169258170654428500644} a^{10} - \frac{247024684234681992143875355}{1150236338516341308857001288} a^{8} - \frac{292995590063717892899426347}{862677253887255981642750966} a^{6} + \frac{230826680647841087096161415}{862677253887255981642750966} a^{4} - \frac{1580632201715195442977327149}{3450709015549023926571003864} a^{2} - \frac{120364707553531269413992709}{287559084629085327214250322}$, $\frac{1}{6901418031098047853142007728} a^{19} - \frac{649811127053677093243197053}{3450709015549023926571003864} a^{17} - \frac{3096878937382613702058360035}{6901418031098047853142007728} a^{15} - \frac{578639135144714617010389583}{6901418031098047853142007728} a^{13} - \frac{282884680093681512375053659}{1150236338516341308857001288} a^{11} + \frac{903211654281659316713125933}{2300472677032682617714002576} a^{9} + \frac{569681663823538088743324619}{1725354507774511963285501932} a^{7} - \frac{631850573239414894546589551}{1725354507774511963285501932} a^{5} - \frac{1580632201715195442977327149}{6901418031098047853142007728} a^{3} - \frac{120364707553531269413992709}{575118169258170654428500644} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 17224568660.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T955:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 819200
The 275 conjugacy class representatives for t20n955 are not computed
Character table for t20n955 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.9932496465625.1

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 R $20$ R ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ $16{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
3Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.0.1$x^{8} + x^{2} - 2 x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.10.8.1$x^{10} - 209 x^{5} + 11552$$5$$2$$8$$D_5$$[\ ]_{5}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$