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Properties

Label 20.0.26769607040...0512.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{4}\cdot 3^{10}\cdot 4903^{5}$
Root discriminant $16.65$
Ramified primes $2, 3, 4903$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1023

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -30, 109, -274, 518, -721, 697, -306, -365, 996, -1177, 770, -76, -484, 685, -563, 322, -135, 42, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 42*x^18 - 135*x^17 + 322*x^16 - 563*x^15 + 685*x^14 - 484*x^13 - 76*x^12 + 770*x^11 - 1177*x^10 + 996*x^9 - 365*x^8 - 306*x^7 + 697*x^6 - 721*x^5 + 518*x^4 - 274*x^3 + 109*x^2 - 30*x + 4)
 
gp: K = bnfinit(x^20 - 9*x^19 + 42*x^18 - 135*x^17 + 322*x^16 - 563*x^15 + 685*x^14 - 484*x^13 - 76*x^12 + 770*x^11 - 1177*x^10 + 996*x^9 - 365*x^8 - 306*x^7 + 697*x^6 - 721*x^5 + 518*x^4 - 274*x^3 + 109*x^2 - 30*x + 4, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 42 x^{18} - 135 x^{17} + 322 x^{16} - 563 x^{15} + 685 x^{14} - 484 x^{13} - 76 x^{12} + 770 x^{11} - 1177 x^{10} + 996 x^{9} - 365 x^{8} - 306 x^{7} + 697 x^{6} - 721 x^{5} + 518 x^{4} - 274 x^{3} + 109 x^{2} - 30 x + 4 \)

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:  \(2676960704021115830230512=2^{4}\cdot 3^{10}\cdot 4903^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.65$
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:  $2, 3, 4903$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{680928060003028} a^{19} - \frac{26325735478709}{170232015000757} a^{18} + \frac{6746070858361}{170232015000757} a^{17} + \frac{289076925254767}{680928060003028} a^{16} + \frac{30558597799719}{680928060003028} a^{15} - \frac{47134819581878}{170232015000757} a^{14} + \frac{11237869930735}{680928060003028} a^{13} - \frac{13806437072363}{40054591764884} a^{12} - \frac{180906020776767}{680928060003028} a^{11} - \frac{183620760836191}{680928060003028} a^{10} + \frac{83742038713036}{170232015000757} a^{9} + \frac{157063832529471}{340464030001514} a^{8} - \frac{311130096638065}{680928060003028} a^{7} + \frac{180830908556531}{680928060003028} a^{6} + \frac{30636601443969}{170232015000757} a^{5} + \frac{37810268756653}{680928060003028} a^{4} + \frac{278477024046217}{680928060003028} a^{3} - \frac{214070293138101}{680928060003028} a^{2} + \frac{77663925323559}{340464030001514} a + \frac{3929280140527}{170232015000757}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{188141149839}{97736193484} a^{19} + \frac{803541325289}{48868096742} a^{18} - \frac{1790553004309}{24434048371} a^{17} + \frac{22089281822871}{97736193484} a^{16} - \frac{50337811582807}{97736193484} a^{15} + \frac{41244530001035}{48868096742} a^{14} - \frac{90258400620509}{97736193484} a^{13} + \frac{2847528105419}{5749187852} a^{12} + \frac{37805307127647}{97736193484} a^{11} - \frac{128109640695637}{97736193484} a^{10} + \frac{80966242871017}{48868096742} a^{9} - \frac{55550545437233}{48868096742} a^{8} + \frac{15418138805667}{97736193484} a^{7} + \frac{66003883041197}{97736193484} a^{6} - \frac{50415090524799}{48868096742} a^{5} + \frac{88320568121921}{97736193484} a^{4} - \frac{55502329626429}{97736193484} a^{3} + \frac{24933185246345}{97736193484} a^{2} - \frac{2081497701360}{24434048371} a + \frac{389119720870}{24434048371} \) (order $6$)
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:  \( 52175.1150233 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.0.5841576387.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: 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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ $20$ $20$ $20$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
3Data not computed
4903Data not computed