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Properties

Label 20.8.65861868411...0144.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{10}\cdot 11^{16}\cdot 241^{3}$
Root discriminant $21.92$
Ramified primes $2, 11, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T747

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 11, -57, 200, -533, 1178, -2143, 2782, -1878, -684, 2420, -1362, -515, 818, -127, -193, 75, 22, -14, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 14*x^18 + 22*x^17 + 75*x^16 - 193*x^15 - 127*x^14 + 818*x^13 - 515*x^12 - 1362*x^11 + 2420*x^10 - 684*x^9 - 1878*x^8 + 2782*x^7 - 2143*x^6 + 1178*x^5 - 533*x^4 + 200*x^3 - 57*x^2 + 11*x - 1)
 
gp: K = bnfinit(x^20 - x^19 - 14*x^18 + 22*x^17 + 75*x^16 - 193*x^15 - 127*x^14 + 818*x^13 - 515*x^12 - 1362*x^11 + 2420*x^10 - 684*x^9 - 1878*x^8 + 2782*x^7 - 2143*x^6 + 1178*x^5 - 533*x^4 + 200*x^3 - 57*x^2 + 11*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 14 x^{18} + 22 x^{17} + 75 x^{16} - 193 x^{15} - 127 x^{14} + 818 x^{13} - 515 x^{12} - 1362 x^{11} + 2420 x^{10} - 684 x^{9} - 1878 x^{8} + 2782 x^{7} - 2143 x^{6} + 1178 x^{5} - 533 x^{4} + 200 x^{3} - 57 x^{2} + 11 x - 1 \)

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:  \(658618684118710741619590144=2^{10}\cdot 11^{16}\cdot 241^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.92$
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:  $2, 11, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3473979948212396237} a^{19} - \frac{671831807094861294}{3473979948212396237} a^{18} + \frac{307202287639760386}{3473979948212396237} a^{17} - \frac{516307138576356137}{3473979948212396237} a^{16} + \frac{159948181598101963}{3473979948212396237} a^{15} - \frac{797804936616541977}{3473979948212396237} a^{14} + \frac{1087727322573461804}{3473979948212396237} a^{13} + \frac{1636563780030478179}{3473979948212396237} a^{12} + \frac{646887523991758449}{3473979948212396237} a^{11} - \frac{1351694384973602997}{3473979948212396237} a^{10} + \frac{700345064364144358}{3473979948212396237} a^{9} + \frac{1165721046073207761}{3473979948212396237} a^{8} + \frac{706070553063777896}{3473979948212396237} a^{7} - \frac{1236055075484155957}{3473979948212396237} a^{6} + \frac{137228531191439063}{3473979948212396237} a^{5} - \frac{1186177752916344627}{3473979948212396237} a^{4} - \frac{871802620041009278}{3473979948212396237} a^{3} + \frac{932157520432483920}{3473979948212396237} a^{2} - \frac{1031989035415288464}{3473979948212396237} a - \frac{739459285035657945}{3473979948212396237}$

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:  \( 616507.960247 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T747:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n747 are not computed
Character table for t20n747 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.51660490321.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ $20$ R $20$ $20$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{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}$
11Data not computed
241Data not computed