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Properties

Label 20.0.15826359080...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 11^{7}\cdot 28162171^{2}$
Root discriminant $57.54$
Ramified primes $2, 5, 11, 28162171$
Class number $8256$ (GRH)
Class group $[2, 4128]$ (GRH)
Galois group 20T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11745679, -2415578, 12643885, -1342832, 6380134, -618928, 2275221, -253640, 586287, -55698, 120498, -13838, 19766, -2000, 2463, -312, 251, -22, 16, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 16*x^18 - 22*x^17 + 251*x^16 - 312*x^15 + 2463*x^14 - 2000*x^13 + 19766*x^12 - 13838*x^11 + 120498*x^10 - 55698*x^9 + 586287*x^8 - 253640*x^7 + 2275221*x^6 - 618928*x^5 + 6380134*x^4 - 1342832*x^3 + 12643885*x^2 - 2415578*x + 11745679)
 
gp: K = bnfinit(x^20 - 2*x^19 + 16*x^18 - 22*x^17 + 251*x^16 - 312*x^15 + 2463*x^14 - 2000*x^13 + 19766*x^12 - 13838*x^11 + 120498*x^10 - 55698*x^9 + 586287*x^8 - 253640*x^7 + 2275221*x^6 - 618928*x^5 + 6380134*x^4 - 1342832*x^3 + 12643885*x^2 - 2415578*x + 11745679, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 16 x^{18} - 22 x^{17} + 251 x^{16} - 312 x^{15} + 2463 x^{14} - 2000 x^{13} + 19766 x^{12} - 13838 x^{11} + 120498 x^{10} - 55698 x^{9} + 586287 x^{8} - 253640 x^{7} + 2275221 x^{6} - 618928 x^{5} + 6380134 x^{4} - 1342832 x^{3} + 12643885 x^{2} - 2415578 x + 11745679 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(158263590809746088460400640000000000=2^{20}\cdot 5^{10}\cdot 11^{7}\cdot 28162171^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.54$
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:  $2, 5, 11, 28162171$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{17} + \frac{1}{3} a^{16} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{106937130679762975418093006160125877058407794290923476549685591} a^{19} + \frac{16130597628863254862789139307367945462651630311706307310904842}{106937130679762975418093006160125877058407794290923476549685591} a^{18} - \frac{33363282729343080462169637510085322024679260780316226663712267}{106937130679762975418093006160125877058407794290923476549685591} a^{17} + \frac{42366549581643850404726909474485044329868738945746311279706706}{106937130679762975418093006160125877058407794290923476549685591} a^{16} - \frac{1867671713253509501392291699888408291568356520948789839654922}{35645710226587658472697668720041959019469264763641158849895197} a^{15} - \frac{18854608813279717663116910131929267054458242720759729637337440}{106937130679762975418093006160125877058407794290923476549685591} a^{14} + \frac{35217658177007806425475846737268105737565088132066940996418994}{106937130679762975418093006160125877058407794290923476549685591} a^{13} + \frac{10566859784615323053142972909845591438129712348584665809430081}{106937130679762975418093006160125877058407794290923476549685591} a^{12} - \frac{35296545501612532183191694474362575181437520458562429243745507}{106937130679762975418093006160125877058407794290923476549685591} a^{11} + \frac{9509165039658563924955880614973931476303138626376402098087615}{35645710226587658472697668720041959019469264763641158849895197} a^{10} - \frac{50957502600541181225309340387672848240314893959761874482521677}{106937130679762975418093006160125877058407794290923476549685591} a^{9} - \frac{38830843026158282545842839713475429409138495157230060063166130}{106937130679762975418093006160125877058407794290923476549685591} a^{8} - \frac{40066149612450374356445719416108757015377409832279288948125234}{106937130679762975418093006160125877058407794290923476549685591} a^{7} + \frac{26374987743905259084805704664928800835237328604376509200566161}{106937130679762975418093006160125877058407794290923476549685591} a^{6} + \frac{36924593931736693339286383668411176089564014357678708258042152}{106937130679762975418093006160125877058407794290923476549685591} a^{5} + \frac{4972383377957045012612823226799287853817624480701258018082416}{106937130679762975418093006160125877058407794290923476549685591} a^{4} + \frac{10257714516963735772908243277012964843123646204681439344652057}{35645710226587658472697668720041959019469264763641158849895197} a^{3} - \frac{52625254894513883415758775367710837483902041102245325262015380}{106937130679762975418093006160125877058407794290923476549685591} a^{2} - \frac{31205488413223883971099078171334218266567170664719993289496930}{106937130679762975418093006160125877058407794290923476549685591} a + \frac{40347239676295117407265767650242850608768222641951509903659384}{106937130679762975418093006160125877058407794290923476549685591}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{4128}$, which has order $8256$ (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:  $9$
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:  \( 170321.970246 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 115200
The 119 conjugacy class representatives for t20n781 are not computed
Character table for t20n781 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.4400.1, 10.10.968074628125.1

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 24 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 R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
28162171Data not computed