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Properties

Label 20.16.2923378425...3201.1
Degree $20$
Signature $[16, 2]$
Discriminant $17^{2}\cdot 97^{2}\cdot 401^{10}$
Root discriminant $42.00$
Ramified primes $17, 97, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T347

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -21, -512, 741, 4788, -1673, -14008, -1049, 18532, 5660, -12644, -5590, 4915, 2464, -1233, -546, 215, 57, -23, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 23*x^18 + 57*x^17 + 215*x^16 - 546*x^15 - 1233*x^14 + 2464*x^13 + 4915*x^12 - 5590*x^11 - 12644*x^10 + 5660*x^9 + 18532*x^8 - 1049*x^7 - 14008*x^6 - 1673*x^5 + 4788*x^4 + 741*x^3 - 512*x^2 - 21*x + 9)
 
gp: K = bnfinit(x^20 - 2*x^19 - 23*x^18 + 57*x^17 + 215*x^16 - 546*x^15 - 1233*x^14 + 2464*x^13 + 4915*x^12 - 5590*x^11 - 12644*x^10 + 5660*x^9 + 18532*x^8 - 1049*x^7 - 14008*x^6 - 1673*x^5 + 4788*x^4 + 741*x^3 - 512*x^2 - 21*x + 9, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 23 x^{18} + 57 x^{17} + 215 x^{16} - 546 x^{15} - 1233 x^{14} + 2464 x^{13} + 4915 x^{12} - 5590 x^{11} - 12644 x^{10} + 5660 x^{9} + 18532 x^{8} - 1049 x^{7} - 14008 x^{6} - 1673 x^{5} + 4788 x^{4} + 741 x^{3} - 512 x^{2} - 21 x + 9 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(292337842510220823284108806723201=17^{2}\cdot 97^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.00$
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:  $17, 97, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{14} + \frac{1}{6} a^{13} - \frac{1}{2} a^{12} + \frac{1}{6} a^{10} + \frac{1}{3} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{16} - \frac{1}{3} a^{13} - \frac{1}{2} a^{12} + \frac{1}{6} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{4} - \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{54} a^{17} - \frac{1}{18} a^{16} + \frac{1}{18} a^{15} - \frac{1}{6} a^{14} - \frac{2}{27} a^{13} - \frac{11}{54} a^{12} + \frac{1}{3} a^{11} - \frac{1}{6} a^{10} + \frac{7}{54} a^{9} + \frac{19}{54} a^{8} + \frac{17}{54} a^{7} - \frac{25}{54} a^{6} - \frac{5}{27} a^{4} + \frac{5}{54} a^{2} - \frac{10}{27} a - \frac{4}{9}$, $\frac{1}{54} a^{18} + \frac{1}{18} a^{16} + \frac{5}{54} a^{14} - \frac{5}{54} a^{13} + \frac{2}{9} a^{12} + \frac{7}{54} a^{10} - \frac{5}{54} a^{9} + \frac{10}{27} a^{8} - \frac{5}{27} a^{7} - \frac{7}{18} a^{6} - \frac{5}{27} a^{5} - \frac{7}{18} a^{4} - \frac{13}{54} a^{3} - \frac{23}{54} a^{2} - \frac{7}{18} a + \frac{1}{6}$, $\frac{1}{27788708345475402677574} a^{19} + \frac{1466310821208881746}{375523085749667603751} a^{18} - \frac{3372645550003201141}{1543817130304189037643} a^{17} + \frac{669518098049314728785}{9262902781825134225858} a^{16} - \frac{700272929327592866654}{13894354172737701338787} a^{15} - \frac{1460549876089929350963}{13894354172737701338787} a^{14} + \frac{3515109943010857109965}{13894354172737701338787} a^{13} - \frac{1466232926586520725575}{9262902781825134225858} a^{12} + \frac{2026604575854237775633}{27788708345475402677574} a^{11} + \frac{2275021622579474420587}{4631451390912567112929} a^{10} + \frac{11450982463517481472093}{27788708345475402677574} a^{9} - \frac{3640375143811772024963}{9262902781825134225858} a^{8} - \frac{4940555392792422897034}{13894354172737701338787} a^{7} + \frac{839605099573744367855}{27788708345475402677574} a^{6} - \frac{11939322463998207463073}{27788708345475402677574} a^{5} - \frac{2561460034868786506561}{27788708345475402677574} a^{4} - \frac{7002840514860611322463}{27788708345475402677574} a^{3} - \frac{3050314246501372845011}{13894354172737701338787} a^{2} - \frac{23903051407801573901}{3087634260608378075286} a - \frac{751318003195735236010}{1543817130304189037643}$

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:  $17$
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:  \( 3879133650.34 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T347:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n347 are not computed
Character table for t20n347 is not computed

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed