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Properties

Label 15.7.25818924764...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{24}\cdot 5^{8}\cdot 13^{10}\cdot 619^{2}\cdot 2731^{2}$
Root discriminant $267.59$
Ramified primes $2, 5, 13, 619, 2731$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T67

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-35195499, 193450599, -134712315, -50194649, 59519321, -506357, -8875651, 1027267, 582272, -96464, -19308, 4336, 261, -93, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 93*x^13 + 261*x^12 + 4336*x^11 - 19308*x^10 - 96464*x^9 + 582272*x^8 + 1027267*x^7 - 8875651*x^6 - 506357*x^5 + 59519321*x^4 - 50194649*x^3 - 134712315*x^2 + 193450599*x - 35195499)
 
gp: K = bnfinit(x^15 - x^14 - 93*x^13 + 261*x^12 + 4336*x^11 - 19308*x^10 - 96464*x^9 + 582272*x^8 + 1027267*x^7 - 8875651*x^6 - 506357*x^5 + 59519321*x^4 - 50194649*x^3 - 134712315*x^2 + 193450599*x - 35195499, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 93 x^{13} + 261 x^{12} + 4336 x^{11} - 19308 x^{10} - 96464 x^{9} + 582272 x^{8} + 1027267 x^{7} - 8875651 x^{6} - 506357 x^{5} + 59519321 x^{4} - 50194649 x^{3} - 134712315 x^{2} + 193450599 x - 35195499 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2581892476484237352007951974400000000=2^{24}\cdot 5^{8}\cdot 13^{10}\cdot 619^{2}\cdot 2731^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $267.59$
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:  $2, 5, 13, 619, 2731$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{18} a^{13} + \frac{5}{18} a^{12} - \frac{1}{9} a^{9} - \frac{1}{3} a^{8} - \frac{1}{9} a^{7} - \frac{2}{9} a^{6} + \frac{1}{18} a^{5} - \frac{7}{18} a^{4} + \frac{2}{9} a^{3} - \frac{2}{9} a^{2} + \frac{1}{18} a - \frac{1}{6}$, $\frac{1}{260544515485650383574986612811916881851633760804} a^{14} - \frac{401043162138909560876055075950968960444067263}{65136128871412595893746653202979220462908440201} a^{13} + \frac{21972649187315337658056801916484083326380456233}{86848171828550127858328870937305627283877920268} a^{12} - \frac{51918082792642737477062231757639366163473826}{804149739153241924614156212382459511887758521} a^{11} + \frac{4686934911385962233743425215617583042584344490}{65136128871412595893746653202979220462908440201} a^{10} - \frac{3342677677386852398828298315012813097816095457}{7237347652379177321527405911442135606989826689} a^{9} + \frac{30112398087739447939758689690526993737345085520}{65136128871412595893746653202979220462908440201} a^{8} + \frac{12605687678231601797967930162712724047518508511}{65136128871412595893746653202979220462908440201} a^{7} - \frac{39670600236563581002283727310906138396652309901}{260544515485650383574986612811916881851633760804} a^{6} + \frac{17706980907562799764381935244325016916233781544}{65136128871412595893746653202979220462908440201} a^{5} - \frac{66979787783365165261494030201242661348590898989}{260544515485650383574986612811916881851633760804} a^{4} - \frac{30162000399952048890319242492006375096090706741}{65136128871412595893746653202979220462908440201} a^{3} - \frac{72756521189952694516072059915698553144563538017}{260544515485650383574986612811916881851633760804} a^{2} - \frac{5994157239712339578663362955420997647294828224}{21712042957137531964582217734326406820969480067} a - \frac{2952369209321383333451644412541001279356875105}{28949390609516709286109623645768542427959306756}$

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:  $10$
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:  \( 8563384625730 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T67:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12000
The 32 conjugacy class representatives for [1/2.F(5)^3]3
Character table for [1/2.F(5)^3]3 is not computed

Intermediate fields

3.3.169.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R $15$ $15$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ $15$ $15$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.24.225$x^{12} - 12 x^{11} + 16 x^{10} - 4 x^{9} - 10 x^{8} + 16 x^{7} - 8 x^{4} - 8 x^{2} + 8$$4$$3$$24$12T55$[2, 2, 3, 3]^{6}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
13Data not computed
619Data not computed
2731Data not computed