↑

Properties

Label 20.10.3698056327...3216.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{10}\cdot 3^{10}\cdot 11^{19}$
Root discriminant $23.90$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T409

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 19, 136, 476, 873, 743, -236, -1291, -799, 767, 649, -433, -175, 193, -23, -49, -2, 9, 5, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 5*x^18 + 9*x^17 - 2*x^16 - 49*x^15 - 23*x^14 + 193*x^13 - 175*x^12 - 433*x^11 + 649*x^10 + 767*x^9 - 799*x^8 - 1291*x^7 - 236*x^6 + 743*x^5 + 873*x^4 + 476*x^3 + 136*x^2 + 19*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 5*x^18 + 9*x^17 - 2*x^16 - 49*x^15 - 23*x^14 + 193*x^13 - 175*x^12 - 433*x^11 + 649*x^10 + 767*x^9 - 799*x^8 - 1291*x^7 - 236*x^6 + 743*x^5 + 873*x^4 + 476*x^3 + 136*x^2 + 19*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 5 x^{18} + 9 x^{17} - 2 x^{16} - 49 x^{15} - 23 x^{14} + 193 x^{13} - 175 x^{12} - 433 x^{11} + 649 x^{10} + 767 x^{9} - 799 x^{8} - 1291 x^{7} - 236 x^{6} + 743 x^{5} + 873 x^{4} + 476 x^{3} + 136 x^{2} + 19 x + 1 \)

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:  \(-3698056327053752876991753216=-\,2^{10}\cdot 3^{10}\cdot 11^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.90$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{461} a^{18} + \frac{220}{461} a^{17} + \frac{223}{461} a^{16} + \frac{154}{461} a^{15} - \frac{34}{461} a^{14} + \frac{153}{461} a^{13} + \frac{141}{461} a^{12} + \frac{79}{461} a^{11} - \frac{27}{461} a^{10} - \frac{187}{461} a^{9} - \frac{229}{461} a^{8} - \frac{165}{461} a^{7} + \frac{176}{461} a^{6} - \frac{3}{461} a^{5} + \frac{94}{461} a^{4} + \frac{91}{461} a^{3} + \frac{223}{461} a^{2} - \frac{113}{461} a - \frac{41}{461}$, $\frac{1}{704175187181471809} a^{19} - \frac{235456384844629}{704175187181471809} a^{18} - \frac{307760441840716566}{704175187181471809} a^{17} - \frac{91316464590910686}{704175187181471809} a^{16} - \frac{148612712564488493}{704175187181471809} a^{15} + \frac{249842768195551461}{704175187181471809} a^{14} - \frac{105201017631215420}{704175187181471809} a^{13} - \frac{13645716104613220}{704175187181471809} a^{12} + \frac{66420907273706074}{704175187181471809} a^{11} - \frac{49008841164065917}{704175187181471809} a^{10} - \frac{120300756145107197}{704175187181471809} a^{9} + \frac{97018569228489233}{704175187181471809} a^{8} - \frac{336085139009250902}{704175187181471809} a^{7} + \frac{180477206743562492}{704175187181471809} a^{6} + \frac{283467335947571593}{704175187181471809} a^{5} + \frac{288512401876028000}{704175187181471809} a^{4} + \frac{301907901382739166}{704175187181471809} a^{3} + \frac{215159606300604929}{704175187181471809} a^{2} - \frac{244039210220816825}{704175187181471809} a + \frac{227766179079251282}{704175187181471809}$

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:  \( 2048882.65605 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T409:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n409 are not computed
Character table for t20n409 is not computed

Intermediate fields

\(\Q(\sqrt{33}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{33})^+\)

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

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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.10.2$x^{10} - 5 x^{8} + 10 x^{6} - 2 x^{4} - 11 x^{2} + 39$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
3Data not computed
11Data not computed