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Properties

Label 20.12.1018381433...0464.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{24}\cdot 13^{15}\cdot 17^{9}$
Root discriminant $56.29$
Ramified primes $2, 13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T803

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -26, 126, 4898, 20511, 4434, -56540, 610, 29636, -16634, 20388, 1606, -6400, -1822, -828, 586, 141, 8, -14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 14*x^18 + 8*x^17 + 141*x^16 + 586*x^15 - 828*x^14 - 1822*x^13 - 6400*x^12 + 1606*x^11 + 20388*x^10 - 16634*x^9 + 29636*x^8 + 610*x^7 - 56540*x^6 + 4434*x^5 + 20511*x^4 + 4898*x^3 + 126*x^2 - 26*x - 1)
 
gp: K = bnfinit(x^20 - 4*x^19 - 14*x^18 + 8*x^17 + 141*x^16 + 586*x^15 - 828*x^14 - 1822*x^13 - 6400*x^12 + 1606*x^11 + 20388*x^10 - 16634*x^9 + 29636*x^8 + 610*x^7 - 56540*x^6 + 4434*x^5 + 20511*x^4 + 4898*x^3 + 126*x^2 - 26*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 14 x^{18} + 8 x^{17} + 141 x^{16} + 586 x^{15} - 828 x^{14} - 1822 x^{13} - 6400 x^{12} + 1606 x^{11} + 20388 x^{10} - 16634 x^{9} + 29636 x^{8} + 610 x^{7} - 56540 x^{6} + 4434 x^{5} + 20511 x^{4} + 4898 x^{3} + 126 x^{2} - 26 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(101838143353046587250933824659390464=2^{24}\cdot 13^{15}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.29$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a$, $\frac{1}{52} a^{18} - \frac{1}{26} a^{17} + \frac{7}{52} a^{16} - \frac{1}{26} a^{15} - \frac{5}{26} a^{13} + \frac{5}{26} a^{12} + \frac{1}{26} a^{11} - \frac{5}{26} a^{10} - \frac{1}{26} a^{9} + \frac{5}{26} a^{8} + \frac{1}{26} a^{7} + \frac{4}{13} a^{6} - \frac{5}{26} a^{5} - \frac{1}{2} a^{4} - \frac{1}{26} a^{3} - \frac{7}{52} a^{2} + \frac{6}{13} a - \frac{1}{52}$, $\frac{1}{26452194096556656335990678399080138704284} a^{19} - \frac{46042306743278002837382186425906268101}{26452194096556656335990678399080138704284} a^{18} + \frac{454010105732568189085707396116847221331}{2034784161273588948922359876852318361868} a^{17} - \frac{1439980180759731048850640185979544877013}{26452194096556656335990678399080138704284} a^{16} - \frac{1684664637604259620262831800940072064591}{13226097048278328167995339199540069352142} a^{15} + \frac{2925952220100767371450582831122413583193}{13226097048278328167995339199540069352142} a^{14} + \frac{423065070313811263384548005203066949381}{13226097048278328167995339199540069352142} a^{13} - \frac{1367255704852219158792378795553028019319}{6613048524139164083997669599770034676071} a^{12} + \frac{3306379946280218800865464649571857905697}{13226097048278328167995339199540069352142} a^{11} - \frac{1558714408359837661458453734390852960646}{6613048524139164083997669599770034676071} a^{10} - \frac{2455381173362756902902017817977004277418}{6613048524139164083997669599770034676071} a^{9} + \frac{2098488906673978605866807797081729648470}{6613048524139164083997669599770034676071} a^{8} - \frac{423497273519720243756889036808224750489}{13226097048278328167995339199540069352142} a^{7} + \frac{968392169190520382554504373728766558155}{13226097048278328167995339199540069352142} a^{6} + \frac{2864464209116324330888736208791254924371}{13226097048278328167995339199540069352142} a^{5} + \frac{2527651410592533412175825986853542661127}{6613048524139164083997669599770034676071} a^{4} + \frac{10379824480527816438652284157266960233801}{26452194096556656335990678399080138704284} a^{3} - \frac{12554741881652539948951135656325324642779}{26452194096556656335990678399080138704284} a^{2} + \frac{12022101262692899261984520180397571202637}{26452194096556656335990678399080138704284} a + \frac{8932044465573311214440577335852782725849}{26452194096556656335990678399080138704284}$

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:  $15$
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:  \( 29862083021.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T803:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 126 conjugacy class representatives for t20n803 are not computed
Character table for t20n803 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.10158928.1, 10.10.1341649635419392.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
2Data not computed
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.12.11.3$x^{12} - 208$$12$$1$$11$$C_{12}$$[\ ]_{12}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.5.1$x^{6} - 17$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$