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Properties

Label 15.15.3801614110...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{24}\cdot 5^{10}\cdot 13^{12}\cdot 315583241^{2}$
Root discriminant $937.56$
Ramified primes $2, 5, 13, 315583241$
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![160028479, 702378508, -278900011, -1656509907, 1074185801, 113400248, -174872823, 9917111, 9834372, -1044166, -226870, 28674, 2167, -296, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 - 296*x^13 + 2167*x^12 + 28674*x^11 - 226870*x^10 - 1044166*x^9 + 9834372*x^8 + 9917111*x^7 - 174872823*x^6 + 113400248*x^5 + 1074185801*x^4 - 1656509907*x^3 - 278900011*x^2 + 702378508*x + 160028479)
 
gp: K = bnfinit(x^15 - 7*x^14 - 296*x^13 + 2167*x^12 + 28674*x^11 - 226870*x^10 - 1044166*x^9 + 9834372*x^8 + 9917111*x^7 - 174872823*x^6 + 113400248*x^5 + 1074185801*x^4 - 1656509907*x^3 - 278900011*x^2 + 702378508*x + 160028479, 1)
 

Normalized defining polynomial

\( x^{15} - 7 x^{14} - 296 x^{13} + 2167 x^{12} + 28674 x^{11} - 226870 x^{10} - 1044166 x^{9} + 9834372 x^{8} + 9917111 x^{7} - 174872823 x^{6} + 113400248 x^{5} + 1074185801 x^{4} - 1656509907 x^{3} - 278900011 x^{2} + 702378508 x + 160028479 \)

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:  \(380161411092019035625018770220810240000000000=2^{24}\cdot 5^{10}\cdot 13^{12}\cdot 315583241^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $937.56$
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:  $2, 5, 13, 315583241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} + \frac{4}{13} a^{11} - \frac{2}{13} a^{10} + \frac{4}{13} a^{9} + \frac{2}{13} a^{8} + \frac{4}{13} a^{7} - \frac{1}{13} a^{6}$, $\frac{1}{13} a^{13} - \frac{5}{13} a^{11} - \frac{1}{13} a^{10} - \frac{1}{13} a^{9} - \frac{4}{13} a^{8} - \frac{4}{13} a^{7} + \frac{4}{13} a^{6}$, $\frac{1}{851192539714119853457180667838171994640526916003126520685} a^{14} + \frac{15874595513005105182141866812141080474585504015102073981}{851192539714119853457180667838171994640526916003126520685} a^{13} + \frac{26288592435605967402127874310893547391143056828717932082}{851192539714119853457180667838171994640526916003126520685} a^{12} + \frac{239428277907955110576853426423412600820258739643141819153}{851192539714119853457180667838171994640526916003126520685} a^{11} + \frac{236307398037347211441869277105578461079324325294201432983}{851192539714119853457180667838171994640526916003126520685} a^{10} + \frac{316393905613818653326764748970060010670021241417244640279}{851192539714119853457180667838171994640526916003126520685} a^{9} + \frac{384904407370315595508536369597927693962446587355644890941}{851192539714119853457180667838171994640526916003126520685} a^{8} - \frac{11816538081662414371857021473168954601100550017012815838}{170238507942823970691436133567634398928105383200625304137} a^{7} + \frac{323091092623456759030468891387800160136401952791923639771}{851192539714119853457180667838171994640526916003126520685} a^{6} + \frac{3132771219621576706602691699930251134140108563345696508}{13095269841755690053187394889818030686777337169278869549} a^{5} + \frac{11352189725617381194914390807333295039467868113792877646}{65476349208778450265936974449090153433886685846394347745} a^{4} - \frac{4197370489719649106622351014488923461813591297524352636}{13095269841755690053187394889818030686777337169278869549} a^{3} - \frac{580965193977123582269268871164120596599554851735587004}{65476349208778450265936974449090153433886685846394347745} a^{2} - \frac{10587266942839703021068823152307624787906974595083016934}{65476349208778450265936974449090153433886685846394347745} a - \frac{4659890056660334858335262057898771457257493534621229151}{65476349208778450265936974449090153433886685846394347745}$

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:  \( 833871897761000000 \) (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 $15$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R $15$ $15$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$

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$$[\ ]^{3}$
2.12.24.231$x^{12} + 24 x^{11} - 26 x^{10} - 20 x^{9} + 6 x^{8} + 24 x^{7} + 16 x^{6} - 16 x^{5} - 4 x^{4} - 16 x^{3} - 8 x^{2} + 16 x - 24$$4$$3$$24$12T55$[2, 2, 3, 3, 3]^{3}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.5.5.2$x^{5} + 5 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
315583241Data not computed