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Properties

Label 20.4.13483238599...4976.6
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 11^{16}\cdot 23^{4}$
Root discriminant $25.50$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T751

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![67, -246, 884, -1710, 1119, 1462, -3829, 1926, 924, -2202, 502, 436, -495, 52, 149, -32, -16, 20, 1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + x^18 + 20*x^17 - 16*x^16 - 32*x^15 + 149*x^14 + 52*x^13 - 495*x^12 + 436*x^11 + 502*x^10 - 2202*x^9 + 924*x^8 + 1926*x^7 - 3829*x^6 + 1462*x^5 + 1119*x^4 - 1710*x^3 + 884*x^2 - 246*x + 67)
 
gp: K = bnfinit(x^20 - 2*x^19 + x^18 + 20*x^17 - 16*x^16 - 32*x^15 + 149*x^14 + 52*x^13 - 495*x^12 + 436*x^11 + 502*x^10 - 2202*x^9 + 924*x^8 + 1926*x^7 - 3829*x^6 + 1462*x^5 + 1119*x^4 - 1710*x^3 + 884*x^2 - 246*x + 67, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + x^{18} + 20 x^{17} - 16 x^{16} - 32 x^{15} + 149 x^{14} + 52 x^{13} - 495 x^{12} + 436 x^{11} + 502 x^{10} - 2202 x^{9} + 924 x^{8} + 1926 x^{7} - 3829 x^{6} + 1462 x^{5} + 1119 x^{4} - 1710 x^{3} + 884 x^{2} - 246 x + 67 \)

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:  \(13483238599952325260241534976=2^{20}\cdot 11^{16}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.50$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{5960624970914967064111151477565421} a^{19} - \frac{2451491183821496060643737445287798}{5960624970914967064111151477565421} a^{18} - \frac{1723805264572210339215288429002352}{5960624970914967064111151477565421} a^{17} - \frac{2735434237534318358221609025087822}{5960624970914967064111151477565421} a^{16} - \frac{1696968520141575656302882408427012}{5960624970914967064111151477565421} a^{15} - \frac{1573007803144385568295302009552742}{5960624970914967064111151477565421} a^{14} - \frac{2675138638017315109456923218124370}{5960624970914967064111151477565421} a^{13} + \frac{2644581110377698639585221974062244}{5960624970914967064111151477565421} a^{12} + \frac{1920592721017953012330064666328973}{5960624970914967064111151477565421} a^{11} - \frac{2917568174330179196418177936992689}{5960624970914967064111151477565421} a^{10} + \frac{2770433845468291410417824937341196}{5960624970914967064111151477565421} a^{9} + \frac{82724097492845340993282007775295}{5960624970914967064111151477565421} a^{8} - \frac{2451168699832756757054159485474832}{5960624970914967064111151477565421} a^{7} + \frac{209253070952981309828958940513938}{5960624970914967064111151477565421} a^{6} + \frac{1938868729379751563904386443169368}{5960624970914967064111151477565421} a^{5} - \frac{491822278225226579343314986079122}{5960624970914967064111151477565421} a^{4} - \frac{2095729653626772663109028983165648}{5960624970914967064111151477565421} a^{3} + \frac{2937622495389183632922403224985199}{5960624970914967064111151477565421} a^{2} - \frac{1038633510126154713470270959946004}{5960624970914967064111151477565421} a + \frac{693139342522508134590802795022529}{5960624970914967064111151477565421}$

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:  \( 1139235.95093 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T751:

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 t20n751 are not computed
Character table for t20n751 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.219503494144.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ 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 ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$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}$
$23$$\Q_{23}$$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}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$