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Properties

Label 15.15.1000663700...7329.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{20}\cdot 251^{2}\cdot 401^{7}\cdot 5227^{2}$
Root discriminant $464.18$
Ramified primes $3, 251, 401, 5227$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T45

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-79330232189952, -99162790237440, -49581395118720, -11710445044896, -864514862676, 179367970671, 40476996000, 928289646, -436778892, -32510241, 1944000, 243000, -3168, -792, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 792*x^13 - 3168*x^12 + 243000*x^11 + 1944000*x^10 - 32510241*x^9 - 436778892*x^8 + 928289646*x^7 + 40476996000*x^6 + 179367970671*x^5 - 864514862676*x^4 - 11710445044896*x^3 - 49581395118720*x^2 - 99162790237440*x - 79330232189952)
 
gp: K = bnfinit(x^15 - 792*x^13 - 3168*x^12 + 243000*x^11 + 1944000*x^10 - 32510241*x^9 - 436778892*x^8 + 928289646*x^7 + 40476996000*x^6 + 179367970671*x^5 - 864514862676*x^4 - 11710445044896*x^3 - 49581395118720*x^2 - 99162790237440*x - 79330232189952, 1)
 

Normalized defining polynomial

\( x^{15} - 792 x^{13} - 3168 x^{12} + 243000 x^{11} + 1944000 x^{10} - 32510241 x^{9} - 436778892 x^{8} + 928289646 x^{7} + 40476996000 x^{6} + 179367970671 x^{5} - 864514862676 x^{4} - 11710445044896 x^{3} - 49581395118720 x^{2} - 99162790237440 x - 79330232189952 \)

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:  \(10006637005875926619966658236409866397329=3^{20}\cdot 251^{2}\cdot 401^{7}\cdot 5227^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $464.18$
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:  $3, 251, 401, 5227$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{9} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{27} a^{5}$, $\frac{1}{162} a^{6} - \frac{1}{18} a^{3} - \frac{1}{2} a$, $\frac{1}{162} a^{7} - \frac{1}{18} a^{4} - \frac{1}{6} a^{2}$, $\frac{1}{486} a^{8} - \frac{1}{54} a^{5} - \frac{1}{18} a^{3}$, $\frac{1}{1458} a^{9} - \frac{1}{18} a^{4} - \frac{1}{18} a^{3} - \frac{1}{2} a$, $\frac{1}{1458} a^{10} - \frac{1}{54} a^{5} - \frac{1}{18} a^{4} - \frac{1}{6} a^{2}$, $\frac{1}{139968} a^{11} - \frac{1}{2916} a^{10} - \frac{1}{5832} a^{9} + \frac{1}{648} a^{7} - \frac{17}{1728} a^{5} + \frac{1}{48} a^{4} - \frac{1}{96} a^{3} + \frac{13}{64} a - \frac{3}{16}$, $\frac{1}{12657586176} a^{12} - \frac{7}{2239488} a^{11} + \frac{3545}{19533312} a^{10} + \frac{281}{2441664} a^{9} - \frac{7481}{19533312} a^{8} - \frac{703}{542592} a^{7} + \frac{3191}{17362944} a^{6} + \frac{235}{45216} a^{5} - \frac{30131}{2893824} a^{4} - \frac{30931}{723456} a^{3} - \frac{308273}{1929216} a^{2} - \frac{70625}{241152} a + \frac{45935}{120576}$, $\frac{1}{810085515264} a^{13} - \frac{1}{101260689408} a^{12} + \frac{39755}{11251187712} a^{11} - \frac{21311}{937598976} a^{10} - \frac{734539}{3750395904} a^{9} - \frac{109}{124416} a^{8} + \frac{804605}{370409472} a^{7} - \frac{30979}{92602368} a^{6} - \frac{914099}{185204736} a^{5} - \frac{468659}{23150592} a^{4} + \frac{20120845}{370409472} a^{3} + \frac{1276781}{30867456} a^{2} - \frac{3157907}{7716864} a + \frac{161999}{1929216}$, $\frac{1}{155536418930688} a^{14} + \frac{5}{12961368244224} a^{13} + \frac{13}{720076013568} a^{12} + \frac{25063}{30003167232} a^{11} - \frac{8547689}{240025337856} a^{10} + \frac{4626887}{60006334464} a^{9} - \frac{151099675}{640067567616} a^{8} - \frac{26239259}{8889827328} a^{7} - \frac{101147873}{106677927936} a^{6} + \frac{122148023}{8889827328} a^{5} + \frac{998644079}{23706206208} a^{4} - \frac{61753853}{1481637888} a^{3} + \frac{76365397}{740818944} a^{2} + \frac{14328577}{30867456} a - \frac{4537465}{10289152}$

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:  \( 2223174963210000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T45:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2430
The 45 conjugacy class representatives for 1/2[3^5:2]D(5)
Character table for 1/2[3^5:2]D(5) is not computed

Intermediate fields

5.5.160801.1

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

Sibling fields

Degree 15 siblings: data not computed
Degree 30 siblings: data not computed
Degree 45 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ R $15$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ $15$ $15$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$3$3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
3.6.8.10$x^{6} + 6 x^{5} + 36$$3$$2$$8$$S_3\times C_3$$[2, 2]^{2}$
3.6.8.4$x^{6} + 18 x^{2} + 63$$3$$2$$8$$C_6$$[2]^{2}$
251Data not computed
401Data not computed
5227Data not computed