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Properties

Label 15.15.2316904364...6192.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 3^{6}\cdot 37^{7}\cdot 83^{6}$
Root discriminant $77.80$
Ramified primes $2, 3, 37, 83$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_5 \times S_3$ (as 15T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9631, 9583, -150660, -135505, 204325, 130210, -125976, -39237, 38628, 2750, -5320, 253, 322, -35, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 - 35*x^13 + 322*x^12 + 253*x^11 - 5320*x^10 + 2750*x^9 + 38628*x^8 - 39237*x^7 - 125976*x^6 + 130210*x^5 + 204325*x^4 - 135505*x^3 - 150660*x^2 + 9583*x + 9631)
 
gp: K = bnfinit(x^15 - 7*x^14 - 35*x^13 + 322*x^12 + 253*x^11 - 5320*x^10 + 2750*x^9 + 38628*x^8 - 39237*x^7 - 125976*x^6 + 130210*x^5 + 204325*x^4 - 135505*x^3 - 150660*x^2 + 9583*x + 9631, 1)
 

Normalized defining polynomial

\( x^{15} - 7 x^{14} - 35 x^{13} + 322 x^{12} + 253 x^{11} - 5320 x^{10} + 2750 x^{9} + 38628 x^{8} - 39237 x^{7} - 125976 x^{6} + 130210 x^{5} + 204325 x^{4} - 135505 x^{3} - 150660 x^{2} + 9583 x + 9631 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(23169043645868181334645896192=2^{10}\cdot 3^{6}\cdot 37^{7}\cdot 83^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.80$
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:  $2, 3, 37, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{15966757187521499571102635837} a^{14} - \frac{5544910509716944655477649941}{15966757187521499571102635837} a^{13} + \frac{1952280339320978729926957558}{15966757187521499571102635837} a^{12} - \frac{3817880760344380699411485964}{15966757187521499571102635837} a^{11} + \frac{2567532151053041246195026387}{15966757187521499571102635837} a^{10} + \frac{59810126451658939328345810}{238309808768977605538845311} a^{9} + \frac{1290610710679424164880413950}{15966757187521499571102635837} a^{8} - \frac{2594651314808748179032401090}{15966757187521499571102635837} a^{7} - \frac{2803694091127052627765794066}{15966757187521499571102635837} a^{6} + \frac{7508992938286945216960146881}{15966757187521499571102635837} a^{5} - \frac{6315400936208396283404594971}{15966757187521499571102635837} a^{4} + \frac{585675973185799922249605697}{15966757187521499571102635837} a^{3} - \frac{3886489102350349926585360914}{15966757187521499571102635837} a^{2} - \frac{5701161480846926314254225469}{15966757187521499571102635837} a + \frac{447931280231517860955630345}{15966757187521499571102635837}$

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:  \( 2117746273.12 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times A_5$ (as 15T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 360
The 15 conjugacy class representatives for $A_5 \times S_3$
Character table for $A_5 \times S_3$

Intermediate fields

3.3.148.1, 5.5.84879369.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 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 R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ $15$ $15$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
2Data not computed
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
37Data not computed
$83$83.6.0.1$x^{6} - x + 34$$1$$6$$0$$C_6$$[\ ]^{6}$
83.9.6.1$x^{9} - 6889 x^{3} + 1715361$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$