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Properties

Label 20.20.2108097156...7677.1
Degree $20$
Signature $[20, 0]$
Discriminant $11^{16}\cdot 43^{5}\cdot 199^{5}$
Root discriminant $65.49$
Ramified primes $11, 43, 199$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T310

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23, -2652, -97022, 473717, -512025, -665992, 1572788, -339422, -1112755, 648233, 287287, -278393, -21081, 55069, -3017, -5652, 651, 292, -43, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 43*x^18 + 292*x^17 + 651*x^16 - 5652*x^15 - 3017*x^14 + 55069*x^13 - 21081*x^12 - 278393*x^11 + 287287*x^10 + 648233*x^9 - 1112755*x^8 - 339422*x^7 + 1572788*x^6 - 665992*x^5 - 512025*x^4 + 473717*x^3 - 97022*x^2 - 2652*x + 23)
 
gp: K = bnfinit(x^20 - 6*x^19 - 43*x^18 + 292*x^17 + 651*x^16 - 5652*x^15 - 3017*x^14 + 55069*x^13 - 21081*x^12 - 278393*x^11 + 287287*x^10 + 648233*x^9 - 1112755*x^8 - 339422*x^7 + 1572788*x^6 - 665992*x^5 - 512025*x^4 + 473717*x^3 - 97022*x^2 - 2652*x + 23, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 43 x^{18} + 292 x^{17} + 651 x^{16} - 5652 x^{15} - 3017 x^{14} + 55069 x^{13} - 21081 x^{12} - 278393 x^{11} + 287287 x^{10} + 648233 x^{9} - 1112755 x^{8} - 339422 x^{7} + 1572788 x^{6} - 665992 x^{5} - 512025 x^{4} + 473717 x^{3} - 97022 x^{2} - 2652 x + 23 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2108097156584902878904709960659367677=11^{16}\cdot 43^{5}\cdot 199^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.49$
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:  $11, 43, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{67} a^{17} + \frac{1}{67} a^{16} + \frac{4}{67} a^{15} + \frac{14}{67} a^{14} + \frac{27}{67} a^{13} + \frac{11}{67} a^{12} - \frac{4}{67} a^{11} - \frac{24}{67} a^{10} - \frac{23}{67} a^{9} - \frac{13}{67} a^{8} - \frac{19}{67} a^{7} + \frac{10}{67} a^{6} - \frac{30}{67} a^{5} - \frac{3}{67} a^{4} - \frac{28}{67} a^{3} + \frac{2}{67} a^{2} - \frac{12}{67} a - \frac{4}{67}$, $\frac{1}{8777} a^{18} - \frac{37}{8777} a^{17} + \frac{2512}{8777} a^{16} + \frac{800}{8777} a^{15} + \frac{768}{8777} a^{14} + \frac{3876}{8777} a^{13} + \frac{2526}{8777} a^{12} - \frac{4294}{8777} a^{11} - \frac{4337}{8777} a^{10} + \frac{2603}{8777} a^{9} + \frac{207}{8777} a^{8} - \frac{1881}{8777} a^{7} - \frac{1616}{8777} a^{6} - \frac{1409}{8777} a^{5} - \frac{3599}{8777} a^{4} - \frac{1480}{8777} a^{3} - \frac{490}{8777} a^{2} + \frac{385}{8777} a - \frac{2528}{8777}$, $\frac{1}{3786439099538957723091526177886377} a^{19} - \frac{87331968536194615920889770068}{3786439099538957723091526177886377} a^{18} - \frac{2786662823830129173085484974952}{3786439099538957723091526177886377} a^{17} - \frac{1291742039065811492199473206645286}{3786439099538957723091526177886377} a^{16} + \frac{1861603390647958016115179996985768}{3786439099538957723091526177886377} a^{15} + \frac{875471220371121508470850352602738}{3786439099538957723091526177886377} a^{14} - \frac{752915853355395937336343182389752}{3786439099538957723091526177886377} a^{13} - \frac{332313955891682292746214203184464}{3786439099538957723091526177886377} a^{12} + \frac{1874364163729759233090490098176671}{3786439099538957723091526177886377} a^{11} + \frac{1063363728177932186045615668024702}{3786439099538957723091526177886377} a^{10} - \frac{1855664470244235356068218655683984}{3786439099538957723091526177886377} a^{9} + \frac{193527575260380807598689382326529}{3786439099538957723091526177886377} a^{8} + \frac{1746621799838662278069386489408685}{3786439099538957723091526177886377} a^{7} + \frac{1304848980606290851520354546624465}{3786439099538957723091526177886377} a^{6} + \frac{567979414935711285629491069012496}{3786439099538957723091526177886377} a^{5} - \frac{1692639671856449930340059699248292}{3786439099538957723091526177886377} a^{4} + \frac{822064412166523633360547538003095}{3786439099538957723091526177886377} a^{3} - \frac{55049301301941576618565581907963}{3786439099538957723091526177886377} a^{2} - \frac{737172130433579843962593248299839}{3786439099538957723091526177886377} a - \frac{594496606067952091252012006184351}{3786439099538957723091526177886377}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  $19$
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:  \( 321368494636 \) (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

\(\Q(\zeta_{11})^+\), 10.10.15695839359943369.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
43Data not computed
199Data not computed