↑

Properties

Label 20.0.15872710287...4704.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 11^{16}\cdot 241^{4}$
Root discriminant $28.84$
Ramified primes $2, 11, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T751

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1979, -6960, 12073, -15276, 17395, -19430, 18849, -15447, 12222, -9616, 6992, -3917, 1444, -223, -8, -71, 110, -69, 25, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 25*x^18 - 69*x^17 + 110*x^16 - 71*x^15 - 8*x^14 - 223*x^13 + 1444*x^12 - 3917*x^11 + 6992*x^10 - 9616*x^9 + 12222*x^8 - 15447*x^7 + 18849*x^6 - 19430*x^5 + 17395*x^4 - 15276*x^3 + 12073*x^2 - 6960*x + 1979)
 
gp: K = bnfinit(x^20 - 6*x^19 + 25*x^18 - 69*x^17 + 110*x^16 - 71*x^15 - 8*x^14 - 223*x^13 + 1444*x^12 - 3917*x^11 + 6992*x^10 - 9616*x^9 + 12222*x^8 - 15447*x^7 + 18849*x^6 - 19430*x^5 + 17395*x^4 - 15276*x^3 + 12073*x^2 - 6960*x + 1979, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 25 x^{18} - 69 x^{17} + 110 x^{16} - 71 x^{15} - 8 x^{14} - 223 x^{13} + 1444 x^{12} - 3917 x^{11} + 6992 x^{10} - 9616 x^{9} + 12222 x^{8} - 15447 x^{7} + 18849 x^{6} - 19430 x^{5} + 17395 x^{4} - 15276 x^{3} + 12073 x^{2} - 6960 x + 1979 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(158727102872609288730321224704=2^{10}\cdot 11^{16}\cdot 241^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.84$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{12} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{13} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{8980322912583578707879289452643773221} a^{19} - \frac{281043968206767639363450783080504029}{2993440970861192902626429817547924407} a^{18} + \frac{376750612017685520265615424383174236}{2993440970861192902626429817547924407} a^{17} + \frac{309748589514852203489966163951689411}{2993440970861192902626429817547924407} a^{16} + \frac{325744812817696758627185872056213714}{2993440970861192902626429817547924407} a^{15} - \frac{1107490901026124218536295170388986368}{8980322912583578707879289452643773221} a^{14} + \frac{234347547833298986904094974915667601}{2993440970861192902626429817547924407} a^{13} - \frac{134543328646418372364872108168834011}{2993440970861192902626429817547924407} a^{12} + \frac{771995937663693125510204292779253913}{8980322912583578707879289452643773221} a^{11} + \frac{318681758066201673496746332422649357}{2993440970861192902626429817547924407} a^{10} + \frac{2695527588407843685153156914420281172}{8980322912583578707879289452643773221} a^{9} + \frac{940657626797364741544706851064269894}{8980322912583578707879289452643773221} a^{8} + \frac{155412192048573803923608885832033403}{2993440970861192902626429817547924407} a^{7} - \frac{4103845465784765285234836410148253480}{8980322912583578707879289452643773221} a^{6} + \frac{1215764640337141751262945065146564712}{8980322912583578707879289452643773221} a^{5} - \frac{4173529333696879082660709309877274989}{8980322912583578707879289452643773221} a^{4} - \frac{536571365873113229746740117494147849}{8980322912583578707879289452643773221} a^{3} - \frac{1131203116283163169836682123297343302}{8980322912583578707879289452643773221} a^{2} + \frac{1360129532648178515238971479085720314}{2993440970861192902626429817547924407} a + \frac{881274793405657191441625107492314038}{2993440970861192902626429817547924407}$

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:  $9$
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:  \( 1755566.60594 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T751:

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 t20n751 are not computed
Character table for t20n751 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
241Data not computed