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Properties

Label 20.10.3958205480...4651.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,11^{17}\cdot 23^{8}$
Root discriminant $26.91$
Ramified primes $11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T310

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, 6, -115, 284, 309, -2541, 4350, -1525, -5047, 8525, -5693, 563, 2181, -2010, 901, -192, -22, 30, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 30*x^18 - 22*x^17 - 192*x^16 + 901*x^15 - 2010*x^14 + 2181*x^13 + 563*x^12 - 5693*x^11 + 8525*x^10 - 5047*x^9 - 1525*x^8 + 4350*x^7 - 2541*x^6 + 309*x^5 + 284*x^4 - 115*x^3 + 6*x^2 + 4*x - 1)
 
gp: K = bnfinit(x^20 - 9*x^19 + 30*x^18 - 22*x^17 - 192*x^16 + 901*x^15 - 2010*x^14 + 2181*x^13 + 563*x^12 - 5693*x^11 + 8525*x^10 - 5047*x^9 - 1525*x^8 + 4350*x^7 - 2541*x^6 + 309*x^5 + 284*x^4 - 115*x^3 + 6*x^2 + 4*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 30 x^{18} - 22 x^{17} - 192 x^{16} + 901 x^{15} - 2010 x^{14} + 2181 x^{13} + 563 x^{12} - 5693 x^{11} + 8525 x^{10} - 5047 x^{9} - 1525 x^{8} + 4350 x^{7} - 2541 x^{6} + 309 x^{5} + 284 x^{4} - 115 x^{3} + 6 x^{2} + 4 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-39582054809133382019675984651=-\,11^{17}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.91$
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:  $11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} + \frac{3}{11} a^{14} + \frac{4}{11} a^{13} + \frac{2}{11} a^{12} + \frac{1}{11} a^{11} - \frac{1}{11} a^{10} + \frac{2}{11} a^{9} - \frac{4}{11} a^{8} - \frac{3}{11} a^{7} - \frac{5}{11} a^{6} - \frac{1}{11} a^{5} + \frac{4}{11} a^{4} + \frac{2}{11} a^{3} + \frac{5}{11} a^{2} - \frac{1}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{16} - \frac{5}{11} a^{14} + \frac{1}{11} a^{13} - \frac{5}{11} a^{12} - \frac{4}{11} a^{11} + \frac{5}{11} a^{10} + \frac{1}{11} a^{9} - \frac{2}{11} a^{8} + \frac{4}{11} a^{7} + \frac{3}{11} a^{6} - \frac{4}{11} a^{5} + \frac{1}{11} a^{4} - \frac{1}{11} a^{3} - \frac{5}{11} a^{2} + \frac{4}{11} a - \frac{3}{11}$, $\frac{1}{11} a^{17} + \frac{5}{11} a^{14} + \frac{4}{11} a^{13} - \frac{5}{11} a^{12} - \frac{1}{11} a^{11} - \frac{4}{11} a^{10} - \frac{3}{11} a^{9} - \frac{5}{11} a^{8} - \frac{1}{11} a^{7} + \frac{4}{11} a^{6} - \frac{4}{11} a^{5} - \frac{3}{11} a^{4} + \frac{5}{11} a^{3} - \frac{4}{11} a^{2} + \frac{3}{11} a + \frac{5}{11}$, $\frac{1}{11} a^{18} - \frac{3}{11} a^{13} + \frac{2}{11} a^{11} + \frac{2}{11} a^{10} - \frac{4}{11} a^{9} - \frac{3}{11} a^{8} - \frac{3}{11} a^{7} - \frac{1}{11} a^{6} + \frac{2}{11} a^{5} - \frac{4}{11} a^{4} - \frac{3}{11} a^{3} - \frac{1}{11} a - \frac{5}{11}$, $\frac{1}{2495724537784402243} a^{19} + \frac{103683148765579925}{2495724537784402243} a^{18} + \frac{6610942121149376}{226884048889491113} a^{17} + \frac{92508445109513408}{2495724537784402243} a^{16} - \frac{58634354145897702}{2495724537784402243} a^{15} - \frac{1234275213749858630}{2495724537784402243} a^{14} - \frac{1209461043694093549}{2495724537784402243} a^{13} - \frac{207974548083242531}{2495724537784402243} a^{12} + \frac{295236589656309578}{2495724537784402243} a^{11} + \frac{740485083607957165}{2495724537784402243} a^{10} - \frac{874630794356739606}{2495724537784402243} a^{9} + \frac{100272614375071301}{226884048889491113} a^{8} - \frac{1189214638791379839}{2495724537784402243} a^{7} - \frac{691607766622587867}{2495724537784402243} a^{6} - \frac{165010449455995584}{2495724537784402243} a^{5} - \frac{844953541886007655}{2495724537784402243} a^{4} + \frac{1089337979628044720}{2495724537784402243} a^{3} + \frac{1131628505018736231}{2495724537784402243} a^{2} - \frac{300662148067488055}{2495724537784402243} a - \frac{85075762010948887}{2495724537784402243}$

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:  $14$
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:  \( 8678971.05833 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T310:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 44 conjugacy class representatives for t20n310
Character table for t20n310 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.113395848049.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$