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Properties

Label 20.2.18256727419...6811.2
Degree $20$
Signature $[2, 9]$
Discriminant $-\,11^{16}\cdot 331^{5}$
Root discriminant $29.04$
Ramified primes $11, 331$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T303

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-241, -3246, -835, -1563, -8743, -6021, -6131, -5621, -219, -2257, 464, -798, 334, -201, 88, -23, 33, -5, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 3*x^18 - 5*x^17 + 33*x^16 - 23*x^15 + 88*x^14 - 201*x^13 + 334*x^12 - 798*x^11 + 464*x^10 - 2257*x^9 - 219*x^8 - 5621*x^7 - 6131*x^6 - 6021*x^5 - 8743*x^4 - 1563*x^3 - 835*x^2 - 3246*x - 241)
 
gp: K = bnfinit(x^20 + 3*x^18 - 5*x^17 + 33*x^16 - 23*x^15 + 88*x^14 - 201*x^13 + 334*x^12 - 798*x^11 + 464*x^10 - 2257*x^9 - 219*x^8 - 5621*x^7 - 6131*x^6 - 6021*x^5 - 8743*x^4 - 1563*x^3 - 835*x^2 - 3246*x - 241, 1)
 

Normalized defining polynomial

\( x^{20} + 3 x^{18} - 5 x^{17} + 33 x^{16} - 23 x^{15} + 88 x^{14} - 201 x^{13} + 334 x^{12} - 798 x^{11} + 464 x^{10} - 2257 x^{9} - 219 x^{8} - 5621 x^{7} - 6131 x^{6} - 6021 x^{5} - 8743 x^{4} - 1563 x^{3} - 835 x^{2} - 3246 x - 241 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-182567274194490055859030886811=-\,11^{16}\cdot 331^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.04$
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:  $11, 331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{76290872923850835191787045048382600393009} a^{19} - \frac{781564539355773372303947811537364896278}{76290872923850835191787045048382600393009} a^{18} - \frac{6818627889668659559737796278506720788714}{76290872923850835191787045048382600393009} a^{17} + \frac{11850313606347947170702379954586424988460}{76290872923850835191787045048382600393009} a^{16} + \frac{19841740351090359187495896173663334807187}{76290872923850835191787045048382600393009} a^{15} + \frac{9980839728715075765309503571538242160578}{76290872923850835191787045048382600393009} a^{14} + \frac{30498361097028202802684121749055276723426}{76290872923850835191787045048382600393009} a^{13} - \frac{1551397303341798876762585741511936649934}{76290872923850835191787045048382600393009} a^{12} + \frac{27234738913434019484586493770542534998626}{76290872923850835191787045048382600393009} a^{11} - \frac{25762250801068266249946121357813210334108}{76290872923850835191787045048382600393009} a^{10} - \frac{22229951363150359253211130954848327718434}{76290872923850835191787045048382600393009} a^{9} + \frac{24266924002510315715105485118024029321469}{76290872923850835191787045048382600393009} a^{8} - \frac{18273882855602065410402895453597612572938}{76290872923850835191787045048382600393009} a^{7} - \frac{37792900844696824776354901233589152070599}{76290872923850835191787045048382600393009} a^{6} + \frac{1040297749129328181468739807082525320434}{76290872923850835191787045048382600393009} a^{5} + \frac{22652881675033652794914970710562560991200}{76290872923850835191787045048382600393009} a^{4} - \frac{3240037086961851790813955409866067970804}{76290872923850835191787045048382600393009} a^{3} + \frac{36831262910249046335449091502369314888234}{76290872923850835191787045048382600393009} a^{2} - \frac{16361880019457197177754205091371766421846}{76290872923850835191787045048382600393009} a - \frac{20514278230246785289583648392373138053556}{76290872923850835191787045048382600393009}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $10$
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:  \( 1032943.16606 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T303:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 50 conjugacy class representatives for t20n303 are not computed
Character table for t20n303 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.70952789611.1

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
Arithmetically equvalently 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 $20$ $20$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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
11Data not computed
331Data not computed