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Properties

Label 20.10.3306757708...2064.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{10}\cdot 11^{18}\cdot 241^{2}$
Root discriminant $21.18$
Ramified primes $2, 11, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T846

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -25, 127, 57, -387, 374, 490, -799, 157, 705, -496, -140, 269, -68, -31, 11, -6, 7, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 3*x^18 + 7*x^17 - 6*x^16 + 11*x^15 - 31*x^14 - 68*x^13 + 269*x^12 - 140*x^11 - 496*x^10 + 705*x^9 + 157*x^8 - 799*x^7 + 490*x^6 + 374*x^5 - 387*x^4 + 57*x^3 + 127*x^2 - 25*x + 1)
 
gp: K = bnfinit(x^20 - x^19 - 3*x^18 + 7*x^17 - 6*x^16 + 11*x^15 - 31*x^14 - 68*x^13 + 269*x^12 - 140*x^11 - 496*x^10 + 705*x^9 + 157*x^8 - 799*x^7 + 490*x^6 + 374*x^5 - 387*x^4 + 57*x^3 + 127*x^2 - 25*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 3 x^{18} + 7 x^{17} - 6 x^{16} + 11 x^{15} - 31 x^{14} - 68 x^{13} + 269 x^{12} - 140 x^{11} - 496 x^{10} + 705 x^{9} + 157 x^{8} - 799 x^{7} + 490 x^{6} + 374 x^{5} - 387 x^{4} + 57 x^{3} + 127 x^{2} - 25 x + 1 \)

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:  \(-330675770864580911767512064=-\,2^{10}\cdot 11^{18}\cdot 241^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.18$
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:  $2, 11, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{16} + \frac{2}{9} a^{14} - \frac{4}{9} a^{13} - \frac{1}{9} a^{12} - \frac{2}{9} a^{11} + \frac{4}{9} a^{9} - \frac{1}{9} a^{8} + \frac{2}{9} a^{7} + \frac{1}{9} a^{6} + \frac{1}{3} a^{5} + \frac{1}{9} a^{4} + \frac{2}{9} a^{3} + \frac{2}{9} a^{2} + \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{27} a^{18} + \frac{1}{27} a^{17} - \frac{2}{27} a^{16} + \frac{2}{27} a^{15} + \frac{1}{3} a^{13} - \frac{13}{27} a^{12} + \frac{5}{27} a^{11} - \frac{5}{27} a^{10} + \frac{7}{27} a^{9} + \frac{1}{3} a^{8} - \frac{13}{27} a^{7} - \frac{13}{27} a^{6} - \frac{2}{27} a^{5} + \frac{13}{27} a^{4} - \frac{1}{9} a^{3} - \frac{1}{27} a^{2} + \frac{4}{27} a + \frac{1}{27}$, $\frac{1}{8235224116466761971} a^{19} - \frac{10598868081224827}{2745074705488920657} a^{18} - \frac{41420279989904495}{2745074705488920657} a^{17} - \frac{667933745043058946}{8235224116466761971} a^{16} - \frac{3111139811256365465}{8235224116466761971} a^{15} + \frac{18570200560703132}{305008300609880073} a^{14} - \frac{200474229003122245}{8235224116466761971} a^{13} + \frac{466925021436086788}{2745074705488920657} a^{12} + \frac{1087256225624270591}{8235224116466761971} a^{11} + \frac{347914651838942263}{915024901829640219} a^{10} + \frac{432033314563671281}{8235224116466761971} a^{9} - \frac{1379728244712481267}{8235224116466761971} a^{8} - \frac{65701771113305414}{2745074705488920657} a^{7} - \frac{1052498193864754828}{8235224116466761971} a^{6} - \frac{432416225461379417}{2745074705488920657} a^{5} - \frac{3855776275141810579}{8235224116466761971} a^{4} - \frac{1037355696644876170}{8235224116466761971} a^{3} + \frac{3439502508759562106}{8235224116466761971} a^{2} - \frac{875523974335691246}{2745074705488920657} a + \frac{2138294149833273332}{8235224116466761971}$

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:  \( 617689.692878 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T846:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 163840
The 649 conjugacy class representatives for t20n846 are not computed
Character table for t20n846 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.51660490321.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$2.10.10.13$x^{10} - 15 x^{8} + 26 x^{6} - 22 x^{4} + 37 x^{2} - 59$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11Data not computed
241Data not computed