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Properties

Label 20.8.12889899207...3296.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{38}\cdot 97^{2}\cdot 2657^{4}$
Root discriminant $28.54$
Ramified primes $2, 97, 2657$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, -40, 188, 436, -1718, -2236, 5858, 4252, -8624, -2420, 6780, -496, -2899, 984, 612, -384, -33, 64, -8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 8*x^18 + 64*x^17 - 33*x^16 - 384*x^15 + 612*x^14 + 984*x^13 - 2899*x^12 - 496*x^11 + 6780*x^10 - 2420*x^9 - 8624*x^8 + 4252*x^7 + 5858*x^6 - 2236*x^5 - 1718*x^4 + 436*x^3 + 188*x^2 - 40*x - 2)
 
gp: K = bnfinit(x^20 - 4*x^19 - 8*x^18 + 64*x^17 - 33*x^16 - 384*x^15 + 612*x^14 + 984*x^13 - 2899*x^12 - 496*x^11 + 6780*x^10 - 2420*x^9 - 8624*x^8 + 4252*x^7 + 5858*x^6 - 2236*x^5 - 1718*x^4 + 436*x^3 + 188*x^2 - 40*x - 2, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 8 x^{18} + 64 x^{17} - 33 x^{16} - 384 x^{15} + 612 x^{14} + 984 x^{13} - 2899 x^{12} - 496 x^{11} + 6780 x^{10} - 2420 x^{9} - 8624 x^{8} + 4252 x^{7} + 5858 x^{6} - 2236 x^{5} - 1718 x^{4} + 436 x^{3} + 188 x^{2} - 40 x - 2 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(128898992075490807544645943296=2^{38}\cdot 97^{2}\cdot 2657^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.54$
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:  $2, 97, 2657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{17} + \frac{1}{4} a^{15} - \frac{1}{2} a^{14} + \frac{1}{4} a^{13} + \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{7799726661130939010949092} a^{19} + \frac{973106602393259696062687}{7799726661130939010949092} a^{18} + \frac{82661348126600841666843}{1949931665282734752737273} a^{17} + \frac{653807471758649272373293}{7799726661130939010949092} a^{16} - \frac{1329950150671258469542577}{3899863330565469505474546} a^{15} - \frac{3446848257569372223178167}{7799726661130939010949092} a^{14} - \frac{2591225721502121262543515}{7799726661130939010949092} a^{13} + \frac{938694454678736980725863}{3899863330565469505474546} a^{12} + \frac{731155453816139452450211}{3899863330565469505474546} a^{11} - \frac{328665917618177666886943}{1949931665282734752737273} a^{10} - \frac{458987354237015803724055}{3899863330565469505474546} a^{9} + \frac{449128207344130072571093}{3899863330565469505474546} a^{8} + \frac{327561466583242615670980}{1949931665282734752737273} a^{7} - \frac{485475031561708749430691}{3899863330565469505474546} a^{6} + \frac{22342948188353261555644}{1949931665282734752737273} a^{5} + \frac{1793610270454571629380803}{3899863330565469505474546} a^{4} + \frac{256436439218874014462223}{1949931665282734752737273} a^{3} + \frac{1674404802927061886970123}{3899863330565469505474546} a^{2} + \frac{1055784083430660114449263}{3899863330565469505474546} a + \frac{371192145139078305479709}{1949931665282734752737273}$

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:  $13$
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:  \( 24856132.1046 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.6.925322313728.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.12.26.64$x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{6} - 2 x^{4} + 4 x^{3} + 2$$12$$1$$26$$S_3 \times C_2^2$$[2, 3]_{3}^{2}$
97Data not computed
2657Data not computed