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Properties

Label 20.6.65861868411...0144.2
Degree $20$
Signature $[6, 7]$
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 20T846

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![353, -808, 1, 1501, -1506, 310, 326, -314, 244, -175, 161, -235, 322, -368, 310, -174, 75, -39, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 21*x^18 - 39*x^17 + 75*x^16 - 174*x^15 + 310*x^14 - 368*x^13 + 322*x^12 - 235*x^11 + 161*x^10 - 175*x^9 + 244*x^8 - 314*x^7 + 326*x^6 + 310*x^5 - 1506*x^4 + 1501*x^3 + x^2 - 808*x + 353)
 
gp: K = bnfinit(x^20 - 7*x^19 + 21*x^18 - 39*x^17 + 75*x^16 - 174*x^15 + 310*x^14 - 368*x^13 + 322*x^12 - 235*x^11 + 161*x^10 - 175*x^9 + 244*x^8 - 314*x^7 + 326*x^6 + 310*x^5 - 1506*x^4 + 1501*x^3 + x^2 - 808*x + 353, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 21 x^{18} - 39 x^{17} + 75 x^{16} - 174 x^{15} + 310 x^{14} - 368 x^{13} + 322 x^{12} - 235 x^{11} + 161 x^{10} - 175 x^{9} + 244 x^{8} - 314 x^{7} + 326 x^{6} + 310 x^{5} - 1506 x^{4} + 1501 x^{3} + x^{2} - 808 x + 353 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{175199477853307396203380359} a^{19} - \frac{85479021415482349469649212}{175199477853307396203380359} a^{18} - \frac{15376183453376256826700781}{175199477853307396203380359} a^{17} - \frac{69362323559343198673672131}{175199477853307396203380359} a^{16} + \frac{8327548738761827237337780}{175199477853307396203380359} a^{15} - \frac{4316505843238915527509162}{175199477853307396203380359} a^{14} - \frac{9927099459697470077705708}{175199477853307396203380359} a^{13} - \frac{13782962875197145707095248}{175199477853307396203380359} a^{12} - \frac{28111277815809487267864934}{175199477853307396203380359} a^{11} - \frac{65281463393022655300381417}{175199477853307396203380359} a^{10} + \frac{376774807542449556029274}{1337400594300056459567789} a^{9} + \frac{22873806420697479425639426}{175199477853307396203380359} a^{8} - \frac{73002665096221477789453159}{175199477853307396203380359} a^{7} + \frac{10513672413918826378583413}{175199477853307396203380359} a^{6} - \frac{13110341318698188705400297}{175199477853307396203380359} a^{5} + \frac{1016149640352479332919833}{175199477853307396203380359} a^{4} - \frac{5059976393476329012880947}{175199477853307396203380359} a^{3} + \frac{79411555059649706572391753}{175199477853307396203380359} a^{2} + \frac{19467747345732639136729940}{175199477853307396203380359} a + \frac{63569406425929417174955222}{175199477853307396203380359}$

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:  $12$
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:  \( 383759.201708 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T846:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 163840
The 649 conjugacy class representatives for t20n846 are not computed
Character table for t20n846 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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$2.5.0.1$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}$
2.10.10.8$x^{10} + x^{8} - 2 x^{6} - 2 x^{4} + x^{2} + 33$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
241Data not computed