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Properties

Label 20.10.2955425558...4411.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,11^{16}\cdot 1451^{5}$
Root discriminant $42.03$
Ramified primes $11, 1451$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T310

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![683, -4245, 7039, 4960, -16808, -10020, 28915, 8528, -32045, 8760, 9973, -8098, 2572, 401, -1045, 483, -78, -4, 11, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 11*x^18 - 4*x^17 - 78*x^16 + 483*x^15 - 1045*x^14 + 401*x^13 + 2572*x^12 - 8098*x^11 + 9973*x^10 + 8760*x^9 - 32045*x^8 + 8528*x^7 + 28915*x^6 - 10020*x^5 - 16808*x^4 + 4960*x^3 + 7039*x^2 - 4245*x + 683)
 
gp: K = bnfinit(x^20 - 6*x^19 + 11*x^18 - 4*x^17 - 78*x^16 + 483*x^15 - 1045*x^14 + 401*x^13 + 2572*x^12 - 8098*x^11 + 9973*x^10 + 8760*x^9 - 32045*x^8 + 8528*x^7 + 28915*x^6 - 10020*x^5 - 16808*x^4 + 4960*x^3 + 7039*x^2 - 4245*x + 683, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 11 x^{18} - 4 x^{17} - 78 x^{16} + 483 x^{15} - 1045 x^{14} + 401 x^{13} + 2572 x^{12} - 8098 x^{11} + 9973 x^{10} + 8760 x^{9} - 32045 x^{8} + 8528 x^{7} + 28915 x^{6} - 10020 x^{5} - 16808 x^{4} + 4960 x^{3} + 7039 x^{2} - 4245 x + 683 \)

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:  \(-295542555810828925413205326764411=-\,11^{16}\cdot 1451^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.03$
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:  $11, 1451$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{67} a^{18} - \frac{21}{67} a^{17} + \frac{32}{67} a^{16} - \frac{5}{67} a^{15} - \frac{31}{67} a^{14} + \frac{6}{67} a^{13} + \frac{6}{67} a^{12} + \frac{21}{67} a^{11} + \frac{24}{67} a^{10} - \frac{26}{67} a^{9} + \frac{24}{67} a^{8} + \frac{31}{67} a^{7} + \frac{31}{67} a^{6} + \frac{21}{67} a^{5} - \frac{11}{67} a^{4} - \frac{16}{67} a^{3} - \frac{1}{67} a^{2} + \frac{31}{67} a - \frac{33}{67}$, $\frac{1}{1864078287613821320291717755592382607} a^{19} + \frac{8495439790271022185517600286234709}{1864078287613821320291717755592382607} a^{18} + \frac{392198129101327994273579831638691752}{1864078287613821320291717755592382607} a^{17} + \frac{503472734380847890922110391890946687}{1864078287613821320291717755592382607} a^{16} + \frac{591517214946795926821129714671104959}{1864078287613821320291717755592382607} a^{15} - \frac{373288719597209737343041969492884270}{1864078287613821320291717755592382607} a^{14} + \frac{284254835764230533530020825332509846}{1864078287613821320291717755592382607} a^{13} + \frac{921507724298539318948958087605274679}{1864078287613821320291717755592382607} a^{12} - \frac{370267956705309217392202654752902385}{1864078287613821320291717755592382607} a^{11} - \frac{189650244782322518255333142061425473}{1864078287613821320291717755592382607} a^{10} - \frac{145176626413730987413157803466571740}{1864078287613821320291717755592382607} a^{9} - \frac{589471193256720796504614415125795677}{1864078287613821320291717755592382607} a^{8} + \frac{532853118562249902070370152258860490}{1864078287613821320291717755592382607} a^{7} - \frac{5272371641158458457963176936071114}{1864078287613821320291717755592382607} a^{6} + \frac{745056240519489815038389166162390687}{1864078287613821320291717755592382607} a^{5} - \frac{15575536283744150804544911734113318}{43350657851484216750970180362613549} a^{4} - \frac{731660647313173202817472763529699716}{1864078287613821320291717755592382607} a^{3} + \frac{364700338832595502059228905375356382}{1864078287613821320291717755592382607} a^{2} - \frac{782188272983410393313581643732279701}{1864078287613821320291717755592382607} a - \frac{815274338127260072187650786014832903}{1864078287613821320291717755592382607}$

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:  \( 613729969.233 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T310:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 44 conjugacy class representatives for t20n310
Character table for t20n310 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.451311402416281.2

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$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}$
1451Data not computed