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Properties

Label 20.12.4877563511...1369.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{2}\cdot 71^{2}\cdot 401^{10}$
Root discriminant $34.23$
Ramified primes $3, 71, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T347

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1109, 4718, 28, -3318, 3038, -1894, -849, -3008, 2190, 2380, -1556, 2038, -2580, -360, 1632, -410, -256, 116, 4, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 4*x^18 + 116*x^17 - 256*x^16 - 410*x^15 + 1632*x^14 - 360*x^13 - 2580*x^12 + 2038*x^11 - 1556*x^10 + 2380*x^9 + 2190*x^8 - 3008*x^7 - 849*x^6 - 1894*x^5 + 3038*x^4 - 3318*x^3 + 28*x^2 + 4718*x - 1109)
 
gp: K = bnfinit(x^20 - 8*x^19 + 4*x^18 + 116*x^17 - 256*x^16 - 410*x^15 + 1632*x^14 - 360*x^13 - 2580*x^12 + 2038*x^11 - 1556*x^10 + 2380*x^9 + 2190*x^8 - 3008*x^7 - 849*x^6 - 1894*x^5 + 3038*x^4 - 3318*x^3 + 28*x^2 + 4718*x - 1109, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 4 x^{18} + 116 x^{17} - 256 x^{16} - 410 x^{15} + 1632 x^{14} - 360 x^{13} - 2580 x^{12} + 2038 x^{11} - 1556 x^{10} + 2380 x^{9} + 2190 x^{8} - 3008 x^{7} - 849 x^{6} - 1894 x^{5} + 3038 x^{4} - 3318 x^{3} + 28 x^{2} + 4718 x - 1109 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4877563511063069089624758321369=3^{2}\cdot 71^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.23$
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:  $3, 71, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{13} - \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{15} + \frac{1}{6} a^{13} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{6} a^{8} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{6} a^{16} + \frac{1}{6} a^{13} - \frac{1}{6} a^{12} + \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{17} + \frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{3} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{18} a^{18} + \frac{1}{18} a^{17} + \frac{1}{18} a^{15} - \frac{1}{18} a^{14} - \frac{1}{9} a^{12} + \frac{1}{6} a^{11} - \frac{2}{9} a^{10} + \frac{2}{9} a^{9} - \frac{2}{9} a^{8} - \frac{1}{9} a^{6} + \frac{1}{18} a^{5} + \frac{5}{18} a^{4} + \frac{1}{18} a^{3} + \frac{4}{9} a - \frac{5}{18}$, $\frac{1}{7621539708590043325072379210106} a^{19} - \frac{39116065832064996017818656859}{3810769854295021662536189605053} a^{18} - \frac{11502289997195611873130021015}{846837745398893702785819912234} a^{17} + \frac{478598374278797111934541919107}{7621539708590043325072379210106} a^{16} - \frac{188004017577551902797253317869}{3810769854295021662536189605053} a^{15} - \frac{2801148791661651737395208887}{149441955070393006373968219806} a^{14} + \frac{6956443679917315729656766423}{224162932605589509560952329709} a^{13} + \frac{69330924111837937563070669728}{423418872699446851392909956117} a^{12} + \frac{293604885189676761851190639761}{7621539708590043325072379210106} a^{11} - \frac{1534347289867017279414070245707}{7621539708590043325072379210106} a^{10} - \frac{70073203289393354872599846865}{224162932605589509560952329709} a^{9} + \frac{137665707158268655556733758}{74720977535196503186984109903} a^{8} - \frac{1263345128648548517579516645461}{3810769854295021662536189605053} a^{7} + \frac{1212861790528056930569226361261}{7621539708590043325072379210106} a^{6} - \frac{2394709743054926626128767689357}{7621539708590043325072379210106} a^{5} - \frac{1537803041767566298614619886005}{3810769854295021662536189605053} a^{4} + \frac{161394411452694867249568571739}{423418872699446851392909956117} a^{3} - \frac{3452982186242492672074525546651}{7621539708590043325072379210106} a^{2} + \frac{3090632137796299412606016215863}{7621539708590043325072379210106} a + \frac{203249790961731724406695292818}{1270256618098340554178729868351}$

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:  $15$
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:  \( 138462893.119 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T347:

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 t20n347 are not computed
Character table for t20n347 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$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}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
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}$
$71$71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
71.4.2.2$x^{4} - 71 x^{2} + 55451$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
71.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
71.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed