↑

Properties

Label 20.2.35028156925...9072.3
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{20}\cdot 11^{16}\cdot 727$
Root discriminant $18.93$
Ramified primes $2, 11, 727$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T749

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, -52, 132, -204, 496, -1235, 1746, -1193, 236, -68, 480, -562, 224, 54, -78, 4, 22, -7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 7*x^18 + 22*x^17 + 4*x^16 - 78*x^15 + 54*x^14 + 224*x^13 - 562*x^12 + 480*x^11 - 68*x^10 + 236*x^9 - 1193*x^8 + 1746*x^7 - 1235*x^6 + 496*x^5 - 204*x^4 + 132*x^3 - 52*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 7*x^18 + 22*x^17 + 4*x^16 - 78*x^15 + 54*x^14 + 224*x^13 - 562*x^12 + 480*x^11 - 68*x^10 + 236*x^9 - 1193*x^8 + 1746*x^7 - 1235*x^6 + 496*x^5 - 204*x^4 + 132*x^3 - 52*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 7 x^{18} + 22 x^{17} + 4 x^{16} - 78 x^{15} + 54 x^{14} + 224 x^{13} - 562 x^{12} + 480 x^{11} - 68 x^{10} + 236 x^{9} - 1193 x^{8} + 1746 x^{7} - 1235 x^{6} + 496 x^{5} - 204 x^{4} + 132 x^{3} - 52 x^{2} + 6 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-35028156925416005746819072=-\,2^{20}\cdot 11^{16}\cdot 727\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.93$
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:  $2, 11, 727$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11} a^{18} - \frac{5}{11} a^{17} + \frac{3}{11} a^{16} + \frac{5}{11} a^{15} - \frac{4}{11} a^{14} - \frac{3}{11} a^{13} - \frac{5}{11} a^{12} + \frac{1}{11} a^{11} - \frac{1}{11} a^{10} + \frac{5}{11} a^{9} - \frac{1}{11} a^{8} + \frac{5}{11} a^{7} - \frac{4}{11} a^{6} - \frac{5}{11} a^{5} - \frac{1}{11} a^{4} - \frac{4}{11} a^{3} - \frac{2}{11} a - \frac{2}{11}$, $\frac{1}{171951940910301563} a^{19} - \frac{7508636480271356}{171951940910301563} a^{18} - \frac{17829734123789473}{171951940910301563} a^{17} - \frac{39013112398860572}{171951940910301563} a^{16} + \frac{31955261137271169}{171951940910301563} a^{15} + \frac{24372774044098637}{171951940910301563} a^{14} + \frac{15665293780916202}{171951940910301563} a^{13} - \frac{44426390525782724}{171951940910301563} a^{12} - \frac{56048586624495154}{171951940910301563} a^{11} - \frac{66524446815906177}{171951940910301563} a^{10} + \frac{11669980305784270}{171951940910301563} a^{9} - \frac{13274294398070848}{171951940910301563} a^{8} - \frac{21006432963034382}{171951940910301563} a^{7} - \frac{7091769369310526}{171951940910301563} a^{6} + \frac{65163792715812781}{171951940910301563} a^{5} + \frac{65794173111333340}{171951940910301563} a^{4} + \frac{46798007098929686}{171951940910301563} a^{3} + \frac{32287841948644786}{171951940910301563} a^{2} + \frac{66806202898913358}{171951940910301563} a - \frac{34482850408938489}{171951940910301563}$

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:  $10$
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:  \( 46087.2944036 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T749:

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 t20n749 are not computed
Character table for t20n749 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.219503494144.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$2.10.10.3$x^{10} - 9 x^{8} + 22 x^{6} - 46 x^{4} + 9 x^{2} - 9$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
2.10.10.3$x^{10} - 9 x^{8} + 22 x^{6} - 46 x^{4} + 9 x^{2} - 9$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
$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}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
727Data not computed