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Properties

Label 20.10.9331576415...1527.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,11^{16}\cdot 727^{5}$
Root discriminant $35.36$
Ramified primes $11, 727$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T303

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1277, -4635, 2515, -6111, -12724, 19476, 2234, -8406, 2526, -364, 682, 933, -1531, -240, 693, -68, -131, 48, 4, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 4*x^18 + 48*x^17 - 131*x^16 - 68*x^15 + 693*x^14 - 240*x^13 - 1531*x^12 + 933*x^11 + 682*x^10 - 364*x^9 + 2526*x^8 - 8406*x^7 + 2234*x^6 + 19476*x^5 - 12724*x^4 - 6111*x^3 + 2515*x^2 - 4635*x + 1277)
 
gp: K = bnfinit(x^20 - 6*x^19 + 4*x^18 + 48*x^17 - 131*x^16 - 68*x^15 + 693*x^14 - 240*x^13 - 1531*x^12 + 933*x^11 + 682*x^10 - 364*x^9 + 2526*x^8 - 8406*x^7 + 2234*x^6 + 19476*x^5 - 12724*x^4 - 6111*x^3 + 2515*x^2 - 4635*x + 1277, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 4 x^{18} + 48 x^{17} - 131 x^{16} - 68 x^{15} + 693 x^{14} - 240 x^{13} - 1531 x^{12} + 933 x^{11} + 682 x^{10} - 364 x^{9} + 2526 x^{8} - 8406 x^{7} + 2234 x^{6} + 19476 x^{5} - 12724 x^{4} - 6111 x^{3} + 2515 x^{2} - 4635 x + 1277 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-9331576415771428587203963681527=-\,11^{16}\cdot 727^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.36$
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:  $11, 727$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{9178899778234699738599537751646170831574591} a^{19} + \frac{3320728074567340195775046838542832092925667}{9178899778234699738599537751646170831574591} a^{18} - \frac{4118489667606907714426865551465124895940505}{9178899778234699738599537751646170831574591} a^{17} + \frac{2254886917498870256706485809596637513700360}{9178899778234699738599537751646170831574591} a^{16} - \frac{4481706632024692712566011189101602424200175}{9178899778234699738599537751646170831574591} a^{15} - \frac{3892401412314519617259420278040449924202528}{9178899778234699738599537751646170831574591} a^{14} - \frac{671847414369637932773016458432865819775512}{9178899778234699738599537751646170831574591} a^{13} + \frac{1462894534074525633744847245070527829206889}{9178899778234699738599537751646170831574591} a^{12} + \frac{2805805944995443762498767389000541645844528}{9178899778234699738599537751646170831574591} a^{11} - \frac{408656338126455392112148131816069573962504}{9178899778234699738599537751646170831574591} a^{10} + \frac{1896166777950859690401797346930334707345459}{9178899778234699738599537751646170831574591} a^{9} + \frac{2553785773408256055567142972456099635733548}{9178899778234699738599537751646170831574591} a^{8} - \frac{207789331081984324508817307482377315810318}{9178899778234699738599537751646170831574591} a^{7} + \frac{197649738126077314259573898182522043595719}{9178899778234699738599537751646170831574591} a^{6} - \frac{1107384951861524346313079899556520586989461}{9178899778234699738599537751646170831574591} a^{5} + \frac{3661602881566772513119992312578988239449987}{9178899778234699738599537751646170831574591} a^{4} + \frac{3206160505870780468164325916540408533488347}{9178899778234699738599537751646170831574591} a^{3} - \frac{95124461722539725198641269230099458137889}{9178899778234699738599537751646170831574591} a^{2} - \frac{4141664889544802137042221895103192381300831}{9178899778234699738599537751646170831574591} a - \frac{2803591727986452697371537734342275935867615}{9178899778234699738599537751646170831574591}$

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:  $14$
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:  \( 86307674.8427 \) (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

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

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
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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
11Data not computed
727Data not computed