Properties

Label 20.0.340...929.1
Degree $20$
Signature $[0, 10]$
Discriminant $3.408\times 10^{21}$
Root discriminant $11.93$
Ramified primes $3, 467, 514417$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group 20T1021

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 2*x^18 - 4*x^17 + 6*x^16 - 4*x^15 + 4*x^14 - 7*x^13 + 5*x^12 - 6*x^11 + 13*x^10 - 14*x^9 + 16*x^8 - 22*x^7 + 21*x^6 - 17*x^5 + 16*x^4 - 11*x^3 + 6*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 2*x^18 - 4*x^17 + 6*x^16 - 4*x^15 + 4*x^14 - 7*x^13 + 5*x^12 - 6*x^11 + 13*x^10 - 14*x^9 + 16*x^8 - 22*x^7 + 21*x^6 - 17*x^5 + 16*x^4 - 11*x^3 + 6*x^2 - 3*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 6, -11, 16, -17, 21, -22, 16, -14, 13, -6, 5, -7, 4, -4, 6, -4, 2, -2, 1]);
 

\( x^{20} - 2 x^{19} + 2 x^{18} - 4 x^{17} + 6 x^{16} - 4 x^{15} + 4 x^{14} - 7 x^{13} + 5 x^{12} - 6 x^{11} + 13 x^{10} - 14 x^{9} + 16 x^{8} - 22 x^{7} + 21 x^{6} - 17 x^{5} + 16 x^{4} - 11 x^{3} + 6 x^{2} - 3 x + 1 \)

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(3407822241034569802929\)\(\medspace = 3^{10}\cdot 467^{2}\cdot 514417^{2}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $11.93$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 467, 514417$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{6197} a^{19} + \frac{117}{6197} a^{18} + \frac{1531}{6197} a^{17} + \frac{2472}{6197} a^{16} + \frac{2915}{6197} a^{15} - \frac{151}{6197} a^{14} + \frac{626}{6197} a^{13} + \frac{123}{6197} a^{12} + \frac{2248}{6197} a^{11} + \frac{1035}{6197} a^{10} - \frac{762}{6197} a^{9} + \frac{2263}{6197} a^{8} + \frac{2842}{6197} a^{7} - \frac{2659}{6197} a^{6} - \frac{353}{6197} a^{5} + \frac{1355}{6197} a^{4} + \frac{139}{6197} a^{3} - \frac{2061}{6197} a^{2} + \frac{2627}{6197} a + \frac{2760}{6197}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{5521}{6197} a^{19} + \frac{10925}{6197} a^{18} - \frac{12337}{6197} a^{17} + \frac{22670}{6197} a^{16} - \frac{31091}{6197} a^{15} + \frac{21864}{6197} a^{14} - \frac{23008}{6197} a^{13} + \frac{33572}{6197} a^{12} - \frac{23405}{6197} a^{11} + \frac{36581}{6197} a^{10} - \frac{68928}{6197} a^{9} + \frac{73493}{6197} a^{8} - \frac{92833}{6197} a^{7} + \frac{117389}{6197} a^{6} - \frac{108491}{6197} a^{5} + \frac{91779}{6197} a^{4} - \frac{79552}{6197} a^{3} + \frac{44468}{6197} a^{2} - \frac{27475}{6197} a + \frac{12857}{6197} \) (order $6$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 658.849285992 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{10}\cdot 658.849285992 \cdot 1}{6\sqrt{3407822241034569802929}}\approx 0.180382767736$ (assuming GRH)

Galois group

20T1021:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 7257600
The 84 conjugacy class representatives for t20n1021 are not computed
Character table for t20n1021 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.0.240232739.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
467Data not computed
514417Data not computed