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Properties

Label 20.8.31427919143...8329.3
Degree $20$
Signature $[8, 6]$
Discriminant $3^{6}\cdot 401^{11}$
Root discriminant $37.57$
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![1291, -254, -5500, 1342, 6651, -3448, -2025, 2468, -634, -112, -273, 188, 378, -550, 161, 182, -155, 22, 18, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 18*x^18 + 22*x^17 - 155*x^16 + 182*x^15 + 161*x^14 - 550*x^13 + 378*x^12 + 188*x^11 - 273*x^10 - 112*x^9 - 634*x^8 + 2468*x^7 - 2025*x^6 - 3448*x^5 + 6651*x^4 + 1342*x^3 - 5500*x^2 - 254*x + 1291)
 
gp: K = bnfinit(x^20 - 8*x^19 + 18*x^18 + 22*x^17 - 155*x^16 + 182*x^15 + 161*x^14 - 550*x^13 + 378*x^12 + 188*x^11 - 273*x^10 - 112*x^9 - 634*x^8 + 2468*x^7 - 2025*x^6 - 3448*x^5 + 6651*x^4 + 1342*x^3 - 5500*x^2 - 254*x + 1291, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 18 x^{18} + 22 x^{17} - 155 x^{16} + 182 x^{15} + 161 x^{14} - 550 x^{13} + 378 x^{12} + 188 x^{11} - 273 x^{10} - 112 x^{9} - 634 x^{8} + 2468 x^{7} - 2025 x^{6} - 3448 x^{5} + 6651 x^{4} + 1342 x^{3} - 5500 x^{2} - 254 x + 1291 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(31427919143590467585816658408329=3^{6}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.57$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{16} + \frac{1}{6} a^{14} + \frac{1}{6} a^{13} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{17} + \frac{1}{6} a^{15} + \frac{1}{6} a^{14} + \frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{3} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a$, $\frac{1}{78} a^{18} - \frac{2}{39} a^{17} + \frac{2}{39} a^{16} - \frac{3}{26} a^{15} - \frac{1}{78} a^{14} + \frac{2}{39} a^{13} + \frac{1}{13} a^{12} + \frac{5}{26} a^{11} - \frac{3}{13} a^{10} + \frac{8}{39} a^{9} - \frac{4}{13} a^{8} - \frac{11}{26} a^{7} - \frac{4}{39} a^{6} - \frac{35}{78} a^{5} - \frac{23}{78} a^{4} + \frac{5}{13} a^{3} + \frac{17}{78} a^{2} - \frac{25}{78} a - \frac{5}{26}$, $\frac{1}{2565369994373000642504118} a^{19} + \frac{729119682865327182682}{1282684997186500321252059} a^{18} - \frac{1901428197812557667653}{427561665728833440417353} a^{17} + \frac{177486347584221207195745}{2565369994373000642504118} a^{16} - \frac{183625074305549924696923}{1282684997186500321252059} a^{15} - \frac{304411478015827641177907}{1282684997186500321252059} a^{14} - \frac{293258286689539904692019}{2565369994373000642504118} a^{13} - \frac{274361112808049912523518}{1282684997186500321252059} a^{12} - \frac{71380044986648968670544}{427561665728833440417353} a^{11} - \frac{204487128051434279984887}{1282684997186500321252059} a^{10} - \frac{59970222805168466709356}{427561665728833440417353} a^{9} + \frac{168215525189729245068437}{2565369994373000642504118} a^{8} - \frac{78341538385083196715968}{1282684997186500321252059} a^{7} + \frac{961833961638236707041157}{2565369994373000642504118} a^{6} - \frac{328079183434069104007069}{855123331457666880834706} a^{5} + \frac{29699294198794772333308}{98668076706653870865543} a^{4} - \frac{441463874923653180274057}{1282684997186500321252059} a^{3} - \frac{21881526383298537235721}{98668076706653870865543} a^{2} - \frac{874249572271526960858621}{2565369994373000642504118} a - \frac{846376432737520287854933}{2565369994373000642504118}$

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:  $13$
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:  \( 155386583.737 \) (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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed