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Properties

Label 20.6.49465059721...4592.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{30}\cdot 7^{11}\cdot 13^{12}$
Root discriminant $38.43$
Ramified primes $2, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T633

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-713, -4184, -8692, -5288, 4839, 2616, -11954, -12216, 2216, 6792, 2168, 1144, 1079, -108, -152, -48, -54, 0, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^18 - 54*x^16 - 48*x^15 - 152*x^14 - 108*x^13 + 1079*x^12 + 1144*x^11 + 2168*x^10 + 6792*x^9 + 2216*x^8 - 12216*x^7 - 11954*x^6 + 2616*x^5 + 4839*x^4 - 5288*x^3 - 8692*x^2 - 4184*x - 713)
 
gp: K = bnfinit(x^20 - 2*x^18 - 54*x^16 - 48*x^15 - 152*x^14 - 108*x^13 + 1079*x^12 + 1144*x^11 + 2168*x^10 + 6792*x^9 + 2216*x^8 - 12216*x^7 - 11954*x^6 + 2616*x^5 + 4839*x^4 - 5288*x^3 - 8692*x^2 - 4184*x - 713, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{18} - 54 x^{16} - 48 x^{15} - 152 x^{14} - 108 x^{13} + 1079 x^{12} + 1144 x^{11} + 2168 x^{10} + 6792 x^{9} + 2216 x^{8} - 12216 x^{7} - 11954 x^{6} + 2616 x^{5} + 4839 x^{4} - 5288 x^{3} - 8692 x^{2} - 4184 x - 713 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-49465059721539005964938842734592=-\,2^{30}\cdot 7^{11}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.43$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{14} a^{15} - \frac{1}{14} a^{14} - \frac{1}{7} a^{13} + \frac{1}{14} a^{12} - \frac{3}{14} a^{11} - \frac{3}{14} a^{10} - \frac{1}{7} a^{9} + \frac{1}{14} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{3}{14} a^{5} + \frac{3}{7} a^{4} - \frac{1}{7} a^{3} + \frac{3}{14} a^{2} + \frac{1}{7}$, $\frac{1}{14} a^{16} - \frac{3}{14} a^{14} - \frac{1}{14} a^{13} - \frac{1}{7} a^{12} + \frac{1}{14} a^{11} + \frac{1}{7} a^{10} - \frac{1}{14} a^{9} - \frac{3}{7} a^{8} - \frac{1}{2} a^{7} + \frac{3}{14} a^{6} - \frac{5}{14} a^{5} + \frac{2}{7} a^{4} - \frac{3}{7} a^{3} - \frac{2}{7} a^{2} - \frac{5}{14} a - \frac{5}{14}$, $\frac{1}{14} a^{17} + \frac{3}{14} a^{14} - \frac{1}{14} a^{13} - \frac{3}{14} a^{12} - \frac{3}{14} a^{10} - \frac{5}{14} a^{9} + \frac{3}{14} a^{8} + \frac{3}{14} a^{7} - \frac{5}{14} a^{6} + \frac{3}{7} a^{5} - \frac{1}{7} a^{4} + \frac{2}{7} a^{3} - \frac{3}{14} a^{2} + \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{1778} a^{18} + \frac{8}{889} a^{17} - \frac{24}{889} a^{16} - \frac{6}{889} a^{15} - \frac{114}{889} a^{14} - \frac{163}{889} a^{13} + \frac{369}{1778} a^{12} - \frac{73}{889} a^{11} - \frac{150}{889} a^{10} + \frac{25}{889} a^{9} + \frac{302}{889} a^{8} + \frac{424}{889} a^{7} - \frac{1}{14} a^{6} + \frac{337}{889} a^{5} - \frac{204}{889} a^{4} + \frac{410}{889} a^{3} - \frac{557}{1778} a^{2} + \frac{398}{889} a - \frac{358}{889}$, $\frac{1}{2322866398192077233283911326} a^{19} + \frac{59937939982841084765899}{331838056884582461897701618} a^{18} + \frac{10481774889982639726714995}{2322866398192077233283911326} a^{17} + \frac{40061817589993485260688698}{1161433199096038616641955663} a^{16} - \frac{2647674971741134457898493}{331838056884582461897701618} a^{15} + \frac{52490801735286630710830216}{1161433199096038616641955663} a^{14} - \frac{158104952979730643479935669}{1161433199096038616641955663} a^{13} + \frac{385286400624199673776247023}{2322866398192077233283911326} a^{12} - \frac{417282334543179744618478003}{2322866398192077233283911326} a^{11} - \frac{177734340736515788157656799}{1161433199096038616641955663} a^{10} - \frac{691487771481118903398521257}{2322866398192077233283911326} a^{9} - \frac{136000784586938810969628804}{1161433199096038616641955663} a^{8} - \frac{658716665437127376121905161}{2322866398192077233283911326} a^{7} + \frac{15455122487221748127042851}{331838056884582461897701618} a^{6} - \frac{94543325752848857135361489}{2322866398192077233283911326} a^{5} - \frac{428335952413406492011929214}{1161433199096038616641955663} a^{4} - \frac{41881124711307954277320937}{2322866398192077233283911326} a^{3} - \frac{5386748629388690385999311}{18290286599937616010109538} a^{2} - \frac{146082427815088827512785348}{1161433199096038616641955663} a + \frac{251877644272952788940914600}{1161433199096038616641955663}$

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:  $12$
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:  \( 228164863.22 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T633:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40960
The 124 conjugacy class representatives for t20n633 are not computed
Character table for t20n633 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.5.6889792.1, 10.10.379753870426112.1

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 $20$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$2$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.6.3$x^{8} - 7 x^{4} + 147$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
13Data not computed