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Properties

Label 20.0.14044288593...6624.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{10}\cdot 881^{8}$
Root discriminant $45.43$
Ramified primes $2, 3, 881$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group 20T277

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1296, 0, -5616, 0, 18000, 0, -21768, 0, 18112, 0, -9260, 0, 3445, 0, -833, 0, 146, 0, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 15*x^18 + 146*x^16 - 833*x^14 + 3445*x^12 - 9260*x^10 + 18112*x^8 - 21768*x^6 + 18000*x^4 - 5616*x^2 + 1296)
 
gp: K = bnfinit(x^20 - 15*x^18 + 146*x^16 - 833*x^14 + 3445*x^12 - 9260*x^10 + 18112*x^8 - 21768*x^6 + 18000*x^4 - 5616*x^2 + 1296, 1)
 

Normalized defining polynomial

\( x^{20} - 15 x^{18} + 146 x^{16} - 833 x^{14} + 3445 x^{12} - 9260 x^{10} + 18112 x^{8} - 21768 x^{6} + 18000 x^{4} - 5616 x^{2} + 1296 \)

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:  \(1404428859331790362883274100506624=2^{16}\cdot 3^{10}\cdot 881^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.43$
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:  $2, 3, 881$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{24} a^{14} - \frac{1}{8} a^{12} + \frac{1}{12} a^{10} - \frac{5}{24} a^{8} - \frac{11}{24} a^{6} + \frac{1}{6} a^{4} + \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{48} a^{15} - \frac{1}{16} a^{13} + \frac{1}{24} a^{11} + \frac{7}{48} a^{9} - \frac{11}{48} a^{7} - \frac{1}{2} a^{6} + \frac{1}{12} a^{5} - \frac{1}{6} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{144} a^{16} - \frac{1}{48} a^{14} + \frac{1}{72} a^{12} - \frac{17}{144} a^{10} + \frac{37}{144} a^{8} - \frac{1}{2} a^{7} + \frac{1}{36} a^{6} + \frac{1}{9} a^{4} + \frac{5}{12} a^{2} - \frac{1}{2} a$, $\frac{1}{864} a^{17} - \frac{1}{288} a^{15} - \frac{1}{48} a^{14} + \frac{19}{432} a^{13} + \frac{1}{16} a^{12} + \frac{163}{864} a^{11} + \frac{5}{24} a^{10} - \frac{35}{864} a^{9} - \frac{19}{48} a^{8} + \frac{25}{54} a^{7} - \frac{13}{48} a^{6} - \frac{95}{216} a^{5} - \frac{1}{3} a^{4} + \frac{5}{72} a^{3} - \frac{1}{12} a^{2} + \frac{1}{3} a - \frac{1}{4}$, $\frac{1}{11606824590048} a^{18} + \frac{8922168841}{3868941530016} a^{16} - \frac{22194855427}{2901706147512} a^{14} - \frac{300965814401}{11606824590048} a^{12} - \frac{1}{4} a^{11} - \frac{1984906192457}{11606824590048} a^{10} - \frac{1}{4} a^{9} - \frac{2819334694585}{5803412295024} a^{8} - \frac{1}{2} a^{7} - \frac{330868665665}{2901706147512} a^{6} + \frac{1}{4} a^{5} + \frac{257772903355}{967235382504} a^{4} - \frac{1}{4} a^{3} - \frac{25329786211}{161205897084} a^{2} - \frac{7929999009}{26867649514}$, $\frac{1}{69640947540288} a^{19} + \frac{8922168841}{23213649180096} a^{17} - \frac{1}{288} a^{16} + \frac{49354783693}{8705118442536} a^{15} - \frac{1}{96} a^{14} - \frac{7555231183181}{69640947540288} a^{13} - \frac{5}{72} a^{12} - \frac{1017670809953}{69640947540288} a^{11} + \frac{41}{288} a^{10} - \frac{6930085070227}{34820473770144} a^{9} - \frac{79}{288} a^{8} + \frac{155111103863}{2176279610634} a^{7} - \frac{23}{144} a^{6} - \frac{2482727347073}{5803412295024} a^{5} + \frac{35}{72} a^{4} + \frac{81115569643}{322411794168} a^{3} - \frac{7}{24} a^{2} - \frac{61665298037}{161205897084} a + \frac{1}{4}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( -\frac{912787517}{2901706147512} a^{18} + \frac{8862925813}{1934470765008} a^{16} - \frac{255515276861}{5803412295024} a^{14} + \frac{708470174329}{2901706147512} a^{12} - \frac{5742678369163}{5803412295024} a^{10} + \frac{14768390073539}{5803412295024} a^{8} - \frac{14043162145991}{2901706147512} a^{6} + \frac{2574424594387}{483617691252} a^{4} - \frac{747791240543}{161205897084} a^{2} + \frac{38752920371}{26867649514} \) (order $6$)
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:  \( 94759599.8631 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T277:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 48 conjugacy class representatives for t20n277
Character table for t20n277 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.3104644.1, 10.10.4163967806429952.2, 10.0.2342231891116848.1, 10.0.1387989268809984.1

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 R R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
881Data not computed