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Properties

Label 20.8.30743450172...2400.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{40}\cdot 5^{2}\cdot 5783^{4}$
Root discriminant $26.57$
Ramified primes $2, 5, 5783$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T794

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17, 84, -92, -160, 336, -60, -94, 248, -519, 288, 334, -432, 79, 88, -144, 0, 49, -4, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 12*x^18 - 4*x^17 + 49*x^16 - 144*x^14 + 88*x^13 + 79*x^12 - 432*x^11 + 334*x^10 + 288*x^9 - 519*x^8 + 248*x^7 - 94*x^6 - 60*x^5 + 336*x^4 - 160*x^3 - 92*x^2 + 84*x + 17)
 
gp: K = bnfinit(x^20 - 12*x^18 - 4*x^17 + 49*x^16 - 144*x^14 + 88*x^13 + 79*x^12 - 432*x^11 + 334*x^10 + 288*x^9 - 519*x^8 + 248*x^7 - 94*x^6 - 60*x^5 + 336*x^4 - 160*x^3 - 92*x^2 + 84*x + 17, 1)
 

Normalized defining polynomial

\( x^{20} - 12 x^{18} - 4 x^{17} + 49 x^{16} - 144 x^{14} + 88 x^{13} + 79 x^{12} - 432 x^{11} + 334 x^{10} + 288 x^{9} - 519 x^{8} + 248 x^{7} - 94 x^{6} - 60 x^{5} + 336 x^{4} - 160 x^{3} - 92 x^{2} + 84 x + 17 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(30743450172982969617507942400=2^{40}\cdot 5^{2}\cdot 5783^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.57$
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:  $2, 5, 5783$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3463} a^{18} + \frac{1174}{3463} a^{17} + \frac{1618}{3463} a^{16} + \frac{1500}{3463} a^{15} + \frac{606}{3463} a^{14} - \frac{1349}{3463} a^{13} - \frac{1666}{3463} a^{12} + \frac{169}{3463} a^{11} + \frac{375}{3463} a^{10} + \frac{1572}{3463} a^{9} + \frac{1095}{3463} a^{8} + \frac{1104}{3463} a^{7} - \frac{370}{3463} a^{6} - \frac{1242}{3463} a^{5} - \frac{77}{3463} a^{4} - \frac{4}{3463} a^{3} - \frac{1585}{3463} a^{2} - \frac{905}{3463} a + \frac{134}{3463}$, $\frac{1}{2282384958465629833774993} a^{19} + \frac{37681274773923209061}{2282384958465629833774993} a^{18} + \frac{506031873632838223437570}{2282384958465629833774993} a^{17} + \frac{659836557187357804721653}{2282384958465629833774993} a^{16} - \frac{860290960660437894866239}{2282384958465629833774993} a^{15} - \frac{623147547171020189069937}{2282384958465629833774993} a^{14} + \frac{505231897590319873920439}{2282384958465629833774993} a^{13} + \frac{913660851284295841505503}{2282384958465629833774993} a^{12} + \frac{111707244827696263292549}{2282384958465629833774993} a^{11} - \frac{749673287650549750537999}{2282384958465629833774993} a^{10} - \frac{43909756418529254250289}{2282384958465629833774993} a^{9} - \frac{413250223775698405181875}{2282384958465629833774993} a^{8} + \frac{880626065707413511778600}{2282384958465629833774993} a^{7} + \frac{358314868696210568008513}{2282384958465629833774993} a^{6} - \frac{598300708073291597505521}{2282384958465629833774993} a^{5} - \frac{121737264487696725037720}{2282384958465629833774993} a^{4} - \frac{933210779613948712163631}{2282384958465629833774993} a^{3} + \frac{1017650162734705766264147}{2282384958465629833774993} a^{2} + \frac{373266521181383629906924}{2282384958465629833774993} a - \frac{98467191454080332763841}{2282384958465629833774993}$

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:  $13$
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:  \( 5621975.60863 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T794:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n794 are not computed
Character table for t20n794 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.3.5783.1, 10.6.1095863140352.2

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.0.1$x^{12} - x^{3} - 2 x + 3$$1$$12$$0$$C_{12}$$[\ ]^{12}$
5783Data not computed