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Properties

Label 23.15.9024061084...9637.1
Degree $23$
Signature $[15, 4]$
Discriminant $9024061084396593245740482274651471479637$
Root discriminant $54.60$
Ramified prime $9024061084396593245740482274651471479637$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_{23}$ (as 23T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -27, -28, 273, 286, -1348, -1386, 3657, 3501, -5930, -4949, 6124, 4062, -4166, -1929, 1869, 490, -531, -45, 86, -5, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 6*x^22 - 5*x^21 + 86*x^20 - 45*x^19 - 531*x^18 + 490*x^17 + 1869*x^16 - 1929*x^15 - 4166*x^14 + 4062*x^13 + 6124*x^12 - 4949*x^11 - 5930*x^10 + 3501*x^9 + 3657*x^8 - 1386*x^7 - 1348*x^6 + 286*x^5 + 273*x^4 - 28*x^3 - 27*x^2 + x + 1)
 
gp: K = bnfinit(x^23 - 6*x^22 - 5*x^21 + 86*x^20 - 45*x^19 - 531*x^18 + 490*x^17 + 1869*x^16 - 1929*x^15 - 4166*x^14 + 4062*x^13 + 6124*x^12 - 4949*x^11 - 5930*x^10 + 3501*x^9 + 3657*x^8 - 1386*x^7 - 1348*x^6 + 286*x^5 + 273*x^4 - 28*x^3 - 27*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{23} - 6 x^{22} - 5 x^{21} + 86 x^{20} - 45 x^{19} - 531 x^{18} + 490 x^{17} + 1869 x^{16} - 1929 x^{15} - 4166 x^{14} + 4062 x^{13} + 6124 x^{12} - 4949 x^{11} - 5930 x^{10} + 3501 x^{9} + 3657 x^{8} - 1386 x^{7} - 1348 x^{6} + 286 x^{5} + 273 x^{4} - 28 x^{3} - 27 x^{2} + x + 1 \)

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

Invariants

Degree:  $23$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9024061084396593245740482274651471479637\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.60$
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:  $9024061084396593245740482274651471479637$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{53} a^{22} + \frac{2}{53} a^{21} + \frac{11}{53} a^{20} + \frac{15}{53} a^{19} + \frac{22}{53} a^{18} + \frac{16}{53} a^{17} - \frac{18}{53} a^{16} - \frac{24}{53} a^{15} - \frac{1}{53} a^{14} + \frac{13}{53} a^{13} - \frac{21}{53} a^{12} + \frac{20}{53} a^{11} - \frac{19}{53} a^{10} + \frac{13}{53} a^{9} + \frac{1}{53} a^{8} + \frac{8}{53} a^{7} + \frac{3}{53} a^{6} + \frac{1}{53} a^{5} - \frac{24}{53} a^{4} - \frac{25}{53} a^{3} - \frac{16}{53} a^{2} + \frac{4}{53} a - \frac{20}{53}$

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:  $18$
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:  \( 1818525117070 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_{23}$ (as 23T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 25852016738884976640000
The 1255 conjugacy class representatives for $S_{23}$ are not computed
Character table for $S_{23}$ 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 ${\href{/LocalNumberField/2.13.0.1}{13} }{,}\,{\href{/LocalNumberField/2.10.0.1}{10} }$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $15{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $17{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ $18{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $23$ $22{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.9.0.1}{9} }$ $23$ $23$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ $17{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
9024061084396593245740482274651471479637Data not computed