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Properties

Label 20.6.18079489015...9712.4
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{10}\cdot 11^{16}\cdot 727^{3}$
Root discriminant $25.87$
Ramified primes $2, 11, 727$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T747

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-43, 102, 401, -595, -1499, 1250, 1387, -2142, 459, 2136, -1128, -228, 1056, -319, -120, 215, -56, -7, 16, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 16*x^18 - 7*x^17 - 56*x^16 + 215*x^15 - 120*x^14 - 319*x^13 + 1056*x^12 - 228*x^11 - 1128*x^10 + 2136*x^9 + 459*x^8 - 2142*x^7 + 1387*x^6 + 1250*x^5 - 1499*x^4 - 595*x^3 + 401*x^2 + 102*x - 43)
 
gp: K = bnfinit(x^20 - 5*x^19 + 16*x^18 - 7*x^17 - 56*x^16 + 215*x^15 - 120*x^14 - 319*x^13 + 1056*x^12 - 228*x^11 - 1128*x^10 + 2136*x^9 + 459*x^8 - 2142*x^7 + 1387*x^6 + 1250*x^5 - 1499*x^4 - 595*x^3 + 401*x^2 + 102*x - 43, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 16 x^{18} - 7 x^{17} - 56 x^{16} + 215 x^{15} - 120 x^{14} - 319 x^{13} + 1056 x^{12} - 228 x^{11} - 1128 x^{10} + 2136 x^{9} + 459 x^{8} - 2142 x^{7} + 1387 x^{6} + 1250 x^{5} - 1499 x^{4} - 595 x^{3} + 401 x^{2} + 102 x - 43 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-18079489015266793067734899712=-\,2^{10}\cdot 11^{16}\cdot 727^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.87$
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:  $2, 11, 727$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} + \frac{4}{23} a^{16} - \frac{10}{23} a^{15} + \frac{7}{23} a^{14} + \frac{2}{23} a^{13} - \frac{10}{23} a^{12} - \frac{2}{23} a^{11} - \frac{7}{23} a^{10} - \frac{9}{23} a^{9} - \frac{8}{23} a^{8} - \frac{4}{23} a^{7} + \frac{6}{23} a^{6} + \frac{6}{23} a^{5} + \frac{11}{23} a^{4} - \frac{9}{23} a^{3} + \frac{3}{23} a^{2} + \frac{11}{23} a + \frac{6}{23}$, $\frac{1}{4710154713115403683649897411} a^{19} + \frac{4843769476533346507629640}{4710154713115403683649897411} a^{18} - \frac{2266195839815535566466512893}{4710154713115403683649897411} a^{17} + \frac{1767225143009879822084822448}{4710154713115403683649897411} a^{16} + \frac{1145002232840926412299758275}{4710154713115403683649897411} a^{15} - \frac{1140609758917052684275661605}{4710154713115403683649897411} a^{14} + \frac{1659950244848237232845811675}{4710154713115403683649897411} a^{13} - \frac{81831956686104438371286369}{204789335352843638419560757} a^{12} + \frac{1898323302627994729070541764}{4710154713115403683649897411} a^{11} + \frac{824441897654038790451559378}{4710154713115403683649897411} a^{10} + \frac{681859303969091230751434752}{4710154713115403683649897411} a^{9} - \frac{1803111920948964781424688498}{4710154713115403683649897411} a^{8} + \frac{2284169886033462111694437420}{4710154713115403683649897411} a^{7} + \frac{2162287914330595357370803013}{4710154713115403683649897411} a^{6} + \frac{1064187151589800940836465245}{4710154713115403683649897411} a^{5} + \frac{1154475169185684945278738616}{4710154713115403683649897411} a^{4} + \frac{1266801054165031838077035861}{4710154713115403683649897411} a^{3} - \frac{1969899820868415079099559342}{4710154713115403683649897411} a^{2} - \frac{1837240348084623747708472825}{4710154713115403683649897411} a + \frac{2278040341151598609614125369}{4710154713115403683649897411}$

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:  $12$
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:  \( 2188619.77218 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T747:

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 t20n747 are not computed
Character table for t20n747 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.155838906487.1

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 $20$ $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.3$x^{10} - 9 x^{8} + 22 x^{6} - 46 x^{4} + 9 x^{2} - 9$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
11Data not computed
727Data not computed