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Properties

Label 23.23.2088046799...0000.1
Degree $23$
Signature $[23, 0]$
Discriminant $2^{22}\cdot 5^{22}\cdot 23^{23}$
Root discriminant $208.09$
Ramified primes $2, 5, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_{23}$ (as 23T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-218280250, -1123046875, 0, 4941406250, 0, -6423828125, 0, 3854296875, 0, -1284765625, 0, 261625000, 0, -34212500, 0, 2932500, 0, -163875, 0, 5750, 0, -115, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 115*x^21 + 5750*x^19 - 163875*x^17 + 2932500*x^15 - 34212500*x^13 + 261625000*x^11 - 1284765625*x^9 + 3854296875*x^7 - 6423828125*x^5 + 4941406250*x^3 - 1123046875*x - 218280250)
 
gp: K = bnfinit(x^23 - 115*x^21 + 5750*x^19 - 163875*x^17 + 2932500*x^15 - 34212500*x^13 + 261625000*x^11 - 1284765625*x^9 + 3854296875*x^7 - 6423828125*x^5 + 4941406250*x^3 - 1123046875*x - 218280250, 1)
 

Normalized defining polynomial

\( x^{23} - 115 x^{21} + 5750 x^{19} - 163875 x^{17} + 2932500 x^{15} - 34212500 x^{13} + 261625000 x^{11} - 1284765625 x^{9} + 3854296875 x^{7} - 6423828125 x^{5} + 4941406250 x^{3} - 1123046875 x - 218280250 \)

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

Invariants

Degree:  $23$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[23, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(208804679998479120343550329105670000000000000000000000=2^{22}\cdot 5^{22}\cdot 23^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $208.09$
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:  $2, 5, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8}$, $\frac{1}{5} a^{9}$, $\frac{1}{5} a^{10}$, $\frac{1}{5} a^{11}$, $\frac{1}{122390} a^{12} - \frac{6923}{122390} a^{11} - \frac{6}{12239} a^{10} - \frac{10883}{122390} a^{9} + \frac{135}{12239} a^{8} - \frac{2712}{12239} a^{7} - \frac{1400}{12239} a^{6} + \frac{10743}{24478} a^{5} - \frac{11353}{24478} a^{4} - \frac{10393}{24478} a^{3} + \frac{989}{12239} a^{2} + \frac{10393}{24478} a + \frac{3125}{12239}$, $\frac{1}{122390} a^{13} - \frac{13}{24478} a^{11} - \frac{10137}{122390} a^{10} + \frac{325}{24478} a^{9} - \frac{3594}{61195} a^{8} - \frac{1950}{12239} a^{7} - \frac{11559}{24478} a^{6} - \frac{864}{12239} a^{5} - \frac{4177}{12239} a^{4} - \frac{7919}{24478} a^{3} - \frac{3593}{24478} a^{2} - \frac{8331}{24478} a - \frac{4177}{12239}$, $\frac{1}{122390} a^{14} + \frac{495}{12239} a^{11} - \frac{455}{24478} a^{10} - \frac{4721}{122390} a^{9} - \frac{2592}{61195} a^{8} + \frac{3051}{24478} a^{7} + \frac{6048}{12239} a^{6} + \frac{4557}{24478} a^{5} - \frac{5762}{12239} a^{4} + \frac{3123}{12239} a^{3} - \frac{2151}{24478} a^{2} + \frac{6285}{24478} a - \frac{4938}{12239}$, $\frac{1}{122390} a^{15} - \frac{525}{24478} a^{11} - \frac{1457}{122390} a^{10} - \frac{5206}{61195} a^{9} - \frac{9229}{122390} a^{8} + \frac{4265}{12239} a^{7} + \frac{10009}{24478} a^{6} + \frac{660}{12239} a^{5} + \frac{1054}{12239} a^{4} - \frac{9557}{24478} a^{3} + \frac{6385}{24478} a^{2} - \frac{1235}{12239} a + \frac{1346}{12239}$, $\frac{1}{611950} a^{16} + \frac{746}{12239} a^{11} + \frac{5239}{61195} a^{10} + \frac{264}{12239} a^{9} + \frac{3709}{61195} a^{8} + \frac{3641}{24478} a^{7} - \frac{528}{12239} a^{6} + \frac{5661}{24478} a^{5} + \frac{2750}{12239} a^{4} + \frac{1773}{12239} a^{3} + \frac{4940}{12239} a^{2} + \frac{3165}{24478} a + \frac{599}{12239}$, $\frac{1}{611950} a^{17} + \frac{3739}{61195} a^{11} + \frac{4818}{61195} a^{10} + \frac{536}{61195} a^{9} + \frac{7663}{122390} a^{8} - \frac{75}{12239} a^{7} - \frac{10551}{24478} a^{6} + \frac{1846}{12239} a^{5} + \frac{1523}{12239} a^{4} - \frac{2322}{12239} a^{3} + \frac{7519}{24478} a^{2} - \frac{4378}{12239} a + \frac{2795}{12239}$, $\frac{1}{611950} a^{18} + \frac{886}{12239} a^{11} + \frac{4574}{61195} a^{10} + \frac{1387}{122390} a^{9} - \frac{5557}{61195} a^{8} - \frac{9925}{24478} a^{7} - \frac{5538}{12239} a^{6} + \frac{1844}{12239} a^{5} + \frac{1693}{12239} a^{4} + \frac{8723}{24478} a^{3} + \frac{4475}{12239} a^{2} + \frac{2193}{12239} a - \frac{4499}{12239}$, $\frac{1}{611950} a^{19} + \frac{506}{12239} a^{11} - \frac{5529}{122390} a^{10} - \frac{3288}{61195} a^{9} + \frac{8073}{122390} a^{8} - \frac{2375}{12239} a^{7} - \frac{4502}{12239} a^{6} - \frac{4565}{12239} a^{5} - \frac{8277}{24478} a^{4} + \frac{2347}{12239} a^{3} + \frac{2777}{12239} a^{2} - \frac{2371}{12239} a - \frac{2882}{12239}$, $\frac{1}{611950} a^{20} - \frac{3167}{122390} a^{11} + \frac{1644}{61195} a^{10} + \frac{553}{122390} a^{9} - \frac{91}{12239} a^{8} - \frac{1701}{12239} a^{7} + \frac{5293}{12239} a^{6} - \frac{2219}{24478} a^{5} + \frac{504}{12239} a^{4} - \frac{4544}{12239} a^{3} - \frac{960}{12239} a^{2} + \frac{4439}{12239} a + \frac{288}{12239}$, $\frac{1}{611950} a^{21} + \frac{2087}{24478} a^{11} + \frac{6357}{122390} a^{10} - \frac{2347}{122390} a^{9} - \frac{366}{61195} a^{8} - \frac{4072}{12239} a^{7} - \frac{8783}{24478} a^{6} - \frac{331}{24478} a^{5} - \frac{5857}{24478} a^{4} + \frac{6359}{24478} a^{3} + \frac{3418}{12239} a^{2} - \frac{7703}{24478} a - \frac{4476}{12239}$, $\frac{1}{611950} a^{22} - \frac{5597}{61195} a^{11} + \frac{11803}{122390} a^{10} + \frac{9931}{122390} a^{9} - \frac{2082}{61195} a^{8} - \frac{2479}{24478} a^{7} - \frac{9063}{24478} a^{6} + \frac{89}{12239} a^{5} + \frac{697}{12239} a^{4} - \frac{4227}{24478} a^{3} + \frac{11299}{24478} a^{2} + \frac{2111}{24478} a - \frac{4679}{12239}$

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:  $22$
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:  \( 108577580716000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_{23}$ (as 23T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 506
The 23 conjugacy class representatives for $F_{23}$
Character table for $F_{23}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 46 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 R $22{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $22{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $22{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $22{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $22{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $22{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
5Data not computed
23Data not computed