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Properties

Label 20.10.8340942746...0523.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,11^{16}\cdot 283^{5}$
Root discriminant $27.93$
Ramified primes $11, 283$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times S_4$ (as 20T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 3, 4, 154, -69, -180, 27, -361, -149, 157, 60, 184, 36, -31, 0, -25, -4, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^17 - 25*x^16 - 31*x^14 + 36*x^13 + 184*x^12 + 60*x^11 + 157*x^10 - 149*x^9 - 361*x^8 + 27*x^7 - 180*x^6 - 69*x^5 + 154*x^4 + 4*x^3 + 3*x^2 - 3*x - 1)
 
gp: K = bnfinit(x^20 - 4*x^17 - 25*x^16 - 31*x^14 + 36*x^13 + 184*x^12 + 60*x^11 + 157*x^10 - 149*x^9 - 361*x^8 + 27*x^7 - 180*x^6 - 69*x^5 + 154*x^4 + 4*x^3 + 3*x^2 - 3*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{17} - 25 x^{16} - 31 x^{14} + 36 x^{13} + 184 x^{12} + 60 x^{11} + 157 x^{10} - 149 x^{9} - 361 x^{8} + 27 x^{7} - 180 x^{6} - 69 x^{5} + 154 x^{4} + 4 x^{3} + 3 x^{2} - 3 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-83409427467164005293746820523=-\,11^{16}\cdot 283^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.93$
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:  $11, 283$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{129597199233731792152259} a^{19} + \frac{64260524753631809201866}{129597199233731792152259} a^{18} - \frac{524790801561143420608}{129597199233731792152259} a^{17} + \frac{44149615588010436558328}{129597199233731792152259} a^{16} + \frac{19666269838438091884625}{129597199233731792152259} a^{15} + \frac{4347615900887221567977}{129597199233731792152259} a^{14} + \frac{43475339553287070238819}{129597199233731792152259} a^{13} + \frac{56309533631213282072063}{129597199233731792152259} a^{12} + \frac{30196276628947875059643}{129597199233731792152259} a^{11} + \frac{56498015992343917379678}{129597199233731792152259} a^{10} - \frac{45044021684239216629071}{129597199233731792152259} a^{9} - \frac{27533157032669692272958}{129597199233731792152259} a^{8} + \frac{41537129658176892739510}{129597199233731792152259} a^{7} - \frac{45307651872176137438002}{129597199233731792152259} a^{6} + \frac{18500430997493630816225}{129597199233731792152259} a^{5} + \frac{61932419812769573170010}{129597199233731792152259} a^{4} - \frac{30692575763025288774099}{129597199233731792152259} a^{3} - \frac{51589499753156682440667}{129597199233731792152259} a^{2} + \frac{6052290635139834324540}{129597199233731792152259} a + \frac{48619691973174426628457}{129597199233731792152259}$

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:  $14$
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:  \( 11612148.1752 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times S_4$ (as 20T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 25 conjugacy class representatives for $C_5\times S_4$
Character table for $C_5\times S_4$ is not computed

Intermediate fields

4.2.283.1, \(\Q(\zeta_{11})^+\)

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

Sibling fields

Degree 30 siblings: 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 $20$ $20$ $20$ $15{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R $15{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{5}$ $15{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ $15{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ $15{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
11Data not computed
283Data not computed