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Properties

Label 20.12.3491991015...6481.3
Degree $20$
Signature $[12, 4]$
Discriminant $3^{4}\cdot 401^{11}$
Root discriminant $33.66$
Ramified primes $3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T350

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![83, -256, -2816, 1056, 5030, -3187, -1310, 1828, -2915, 1900, 2146, -1326, -157, -213, -107, 151, 5, 14, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 6*x^18 + 14*x^17 + 5*x^16 + 151*x^15 - 107*x^14 - 213*x^13 - 157*x^12 - 1326*x^11 + 2146*x^10 + 1900*x^9 - 2915*x^8 + 1828*x^7 - 1310*x^6 - 3187*x^5 + 5030*x^4 + 1056*x^3 - 2816*x^2 - 256*x + 83)
 
gp: K = bnfinit(x^20 - 4*x^19 - 6*x^18 + 14*x^17 + 5*x^16 + 151*x^15 - 107*x^14 - 213*x^13 - 157*x^12 - 1326*x^11 + 2146*x^10 + 1900*x^9 - 2915*x^8 + 1828*x^7 - 1310*x^6 - 3187*x^5 + 5030*x^4 + 1056*x^3 - 2816*x^2 - 256*x + 83, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 6 x^{18} + 14 x^{17} + 5 x^{16} + 151 x^{15} - 107 x^{14} - 213 x^{13} - 157 x^{12} - 1326 x^{11} + 2146 x^{10} + 1900 x^{9} - 2915 x^{8} + 1828 x^{7} - 1310 x^{6} - 3187 x^{5} + 5030 x^{4} + 1056 x^{3} - 2816 x^{2} - 256 x + 83 \)

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:  \(3491991015954496398424073156481=3^{4}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.66$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{117} a^{18} + \frac{16}{117} a^{17} - \frac{5}{39} a^{16} - \frac{4}{39} a^{15} - \frac{19}{117} a^{14} + \frac{14}{117} a^{13} - \frac{50}{117} a^{12} + \frac{31}{117} a^{11} - \frac{4}{9} a^{10} - \frac{7}{117} a^{9} + \frac{4}{117} a^{8} - \frac{46}{117} a^{7} - \frac{1}{39} a^{6} - \frac{8}{39} a^{5} + \frac{16}{117} a^{4} - \frac{41}{117} a^{3} - \frac{40}{117} a^{2} - \frac{22}{117} a - \frac{41}{117}$, $\frac{1}{80341142306484525664612887873918531} a^{19} - \frac{72317595363247891503083618143669}{26780380768828175221537629291306177} a^{18} - \frac{187168512053297680674454743324277}{80341142306484525664612887873918531} a^{17} + \frac{213203425962411667318685913248327}{1575316515813422071855154664194481} a^{16} - \frac{904509100120336961004611876574751}{6180087869729578897277914451839887} a^{15} - \frac{69300278003160672230176232544859}{26780380768828175221537629291306177} a^{14} - \frac{26395324700949309270800829419127388}{80341142306484525664612887873918531} a^{13} - \frac{3564881134876908355646741133706515}{8926793589609391740512543097102059} a^{12} + \frac{17999679071142252936716671479678064}{80341142306484525664612887873918531} a^{11} - \frac{1156854466223083689123439616043722}{8926793589609391740512543097102059} a^{10} - \frac{29720597442359852171859295673406481}{80341142306484525664612887873918531} a^{9} - \frac{26739255894242507659986340901660582}{80341142306484525664612887873918531} a^{8} + \frac{31932263241037787722133344248718717}{80341142306484525664612887873918531} a^{7} - \frac{1948020662722162142476886226827827}{26780380768828175221537629291306177} a^{6} + \frac{20463469862618977788018334090483174}{80341142306484525664612887873918531} a^{5} - \frac{537817582205847383660218662514330}{26780380768828175221537629291306177} a^{4} + \frac{2216280056087120679958309255244464}{6180087869729578897277914451839887} a^{3} - \frac{110835272181609498312513325195847}{8926793589609391740512543097102059} a^{2} - \frac{3838716275955541244910114692286223}{80341142306484525664612887873918531} a + \frac{316620498710456715829931826659458}{967965569957644887525456480408657}$

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:  \( 155213780.152 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T350:

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 t20n350 are not computed
Character table for t20n350 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed