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Properties

Label 20.4.48982150162...2449.1
Degree $20$
Signature $[4, 8]$
Discriminant $11^{16}\cdot 23^{2}\cdot 67^{4}$
Root discriminant $21.60$
Ramified primes $11, 23, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11, 0, 0, 11, 77, -23, -119, 152, 529, 106, -682, -331, 197, 250, -115, -40, 85, -63, 26, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 26*x^18 - 63*x^17 + 85*x^16 - 40*x^15 - 115*x^14 + 250*x^13 + 197*x^12 - 331*x^11 - 682*x^10 + 106*x^9 + 529*x^8 + 152*x^7 - 119*x^6 - 23*x^5 + 77*x^4 + 11*x^3 - 11)
 
gp: K = bnfinit(x^20 - 7*x^19 + 26*x^18 - 63*x^17 + 85*x^16 - 40*x^15 - 115*x^14 + 250*x^13 + 197*x^12 - 331*x^11 - 682*x^10 + 106*x^9 + 529*x^8 + 152*x^7 - 119*x^6 - 23*x^5 + 77*x^4 + 11*x^3 - 11, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 26 x^{18} - 63 x^{17} + 85 x^{16} - 40 x^{15} - 115 x^{14} + 250 x^{13} + 197 x^{12} - 331 x^{11} - 682 x^{10} + 106 x^{9} + 529 x^{8} + 152 x^{7} - 119 x^{6} - 23 x^{5} + 77 x^{4} + 11 x^{3} - 11 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(489821501624624581418972449=11^{16}\cdot 23^{2}\cdot 67^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.60$
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:  $11, 23, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{4059993703291368646910608171} a^{19} + \frac{600401298569759719439877076}{4059993703291368646910608171} a^{18} + \frac{344369166743506758016307792}{4059993703291368646910608171} a^{17} - \frac{807396131543200966197039611}{4059993703291368646910608171} a^{16} + \frac{888304295653850225689059263}{4059993703291368646910608171} a^{15} + \frac{442290725353471011711800855}{4059993703291368646910608171} a^{14} + \frac{97518971362948667845938010}{4059993703291368646910608171} a^{13} - \frac{48364050085935761441606331}{4059993703291368646910608171} a^{12} - \frac{363304457093860961621312237}{4059993703291368646910608171} a^{11} + \frac{723653816939219550913853584}{4059993703291368646910608171} a^{10} - \frac{2018058416284987105298854979}{4059993703291368646910608171} a^{9} + \frac{957841593498995527270890569}{4059993703291368646910608171} a^{8} - \frac{1734591548692103063360359643}{4059993703291368646910608171} a^{7} - \frac{1358615414674229688015015021}{4059993703291368646910608171} a^{6} + \frac{1248826768250770824504626031}{4059993703291368646910608171} a^{5} - \frac{1845225976065178579503728160}{4059993703291368646910608171} a^{4} + \frac{510826320910976739336008589}{4059993703291368646910608171} a^{3} + \frac{23596495420271105024369216}{4059993703291368646910608171} a^{2} + \frac{1248245762451226444089652531}{4059993703291368646910608171} a - \frac{1754413019754648344498270115}{4059993703291368646910608171}$

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:  $11$
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:  \( 312451.754252 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2560
The 112 conjugacy class representatives for t20n263 are not computed
Character table for t20n263 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.330327035621.1, 10.6.330327035621.4, 10.2.962257016809.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$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}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$