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Properties

Label 20.12.2591155743...5521.2
Degree $20$
Signature $[12, 4]$
Discriminant $11^{16}\cdot 23^{4}\cdot 67^{4}$
Root discriminant $29.56$
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![-253, 902, 1705, -3113, -3806, 155, 1398, 4356, 736, -633, -136, -2676, 473, 649, -115, 194, -85, -3, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 3*x^17 - 85*x^16 + 194*x^15 - 115*x^14 + 649*x^13 + 473*x^12 - 2676*x^11 - 136*x^10 - 633*x^9 + 736*x^8 + 4356*x^7 + 1398*x^6 + 155*x^5 - 3806*x^4 - 3113*x^3 + 1705*x^2 + 902*x - 253)
 
gp: K = bnfinit(x^20 - 2*x^19 - 3*x^17 - 85*x^16 + 194*x^15 - 115*x^14 + 649*x^13 + 473*x^12 - 2676*x^11 - 136*x^10 - 633*x^9 + 736*x^8 + 4356*x^7 + 1398*x^6 + 155*x^5 - 3806*x^4 - 3113*x^3 + 1705*x^2 + 902*x - 253, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 3 x^{17} - 85 x^{16} + 194 x^{15} - 115 x^{14} + 649 x^{13} + 473 x^{12} - 2676 x^{11} - 136 x^{10} - 633 x^{9} + 736 x^{8} + 4356 x^{7} + 1398 x^{6} + 155 x^{5} - 3806 x^{4} - 3113 x^{3} + 1705 x^{2} + 902 x - 253 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(259115574359426403570636425521=11^{16}\cdot 23^{4}\cdot 67^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.56$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11} a^{18} + \frac{3}{11} a^{17} - \frac{5}{11} a^{16} + \frac{4}{11} a^{14} + \frac{5}{11} a^{13} - \frac{5}{11} a^{12} - \frac{4}{11} a^{11} + \frac{3}{11} a^{10} + \frac{4}{11} a^{9} + \frac{2}{11} a^{7} - \frac{2}{11} a^{6} + \frac{5}{11} a^{5}$, $\frac{1}{40763991812368886036169936173653633709} a^{19} - \frac{1371245185253143048930870504043042292}{40763991812368886036169936173653633709} a^{18} - \frac{15000806054897896043639340418100944606}{40763991812368886036169936173653633709} a^{17} + \frac{16877106238925358442813137202931200792}{40763991812368886036169936173653633709} a^{16} + \frac{14510478673254746086590531234715912458}{40763991812368886036169936173653633709} a^{15} - \frac{6560298457817701340872360847142074809}{40763991812368886036169936173653633709} a^{14} - \frac{7168975001960704932549801699436009940}{40763991812368886036169936173653633709} a^{13} + \frac{10509562123980805979131332443302885124}{40763991812368886036169936173653633709} a^{12} - \frac{13011004412777175219410658313267116681}{40763991812368886036169936173653633709} a^{11} - \frac{18151969841140819475157212685030583062}{40763991812368886036169936173653633709} a^{10} + \frac{14575537542290565775747079273125467125}{40763991812368886036169936173653633709} a^{9} + \frac{20359613942666895874455774106984819142}{40763991812368886036169936173653633709} a^{8} - \frac{10991880355934235102833678179691875754}{40763991812368886036169936173653633709} a^{7} + \frac{5885332525141965889192151943103847394}{40763991812368886036169936173653633709} a^{6} + \frac{3779607469772879687022105646206913876}{40763991812368886036169936173653633709} a^{5} - \frac{1539098750346788692967938023224773397}{3705817437488080548742721470332148519} a^{4} + \frac{291215999199425020757403519621491896}{3705817437488080548742721470332148519} a^{3} - \frac{1192331164755945860854363149312517808}{3705817437488080548742721470332148519} a^{2} + \frac{406826049189897172120822072820521324}{3705817437488080548742721470332148519} a + \frac{159749259543289516871518786358479318}{3705817437488080548742721470332148519}$

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:  $15$
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:  \( 39378334.8224 \) (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

\(\Q(\zeta_{11})^+\), 10.6.330327035621.2, 10.6.330327035621.3, 10.10.509033961891961.1

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 ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$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}$
67Data not computed