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Properties

Label 15.7.20785321322...6528.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{24}\cdot 3^{20}\cdot 19^{2}\cdot 73^{3}\cdot 503^{2}$
Root discriminant $105.00$
Ramified primes $2, 3, 19, 73, 503$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T75

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![90953, 103137, -219972, -402517, -205428, -44982, 13036, 32784, -1359, -15815, -3972, 927, 237, -45, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 - 45*x^13 + 237*x^12 + 927*x^11 - 3972*x^10 - 15815*x^9 - 1359*x^8 + 32784*x^7 + 13036*x^6 - 44982*x^5 - 205428*x^4 - 402517*x^3 - 219972*x^2 + 103137*x + 90953)
 
gp: K = bnfinit(x^15 - 6*x^14 - 45*x^13 + 237*x^12 + 927*x^11 - 3972*x^10 - 15815*x^9 - 1359*x^8 + 32784*x^7 + 13036*x^6 - 44982*x^5 - 205428*x^4 - 402517*x^3 - 219972*x^2 + 103137*x + 90953, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{14} - 45 x^{13} + 237 x^{12} + 927 x^{11} - 3972 x^{10} - 15815 x^{9} - 1359 x^{8} + 32784 x^{7} + 13036 x^{6} - 44982 x^{5} - 205428 x^{4} - 402517 x^{3} - 219972 x^{2} + 103137 x + 90953 \)

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:  \(2078532132292195957110397206528=2^{24}\cdot 3^{20}\cdot 19^{2}\cdot 73^{3}\cdot 503^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.00$
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:  $2, 3, 19, 73, 503$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} + \frac{1}{11} a^{11} + \frac{1}{11} a^{10} + \frac{1}{11} a^{8} - \frac{2}{11} a^{7} - \frac{5}{11} a^{6} + \frac{3}{11} a^{5} - \frac{2}{11} a^{4} + \frac{2}{11} a^{3} + \frac{5}{11} a^{2} - \frac{4}{11} a + \frac{3}{11}$, $\frac{1}{4166324251388991064948609718443166143} a^{14} + \frac{52568514011599058888980009261398931}{4166324251388991064948609718443166143} a^{13} + \frac{2081067674519491354820752501625574867}{4166324251388991064948609718443166143} a^{12} + \frac{161820780484675004944378256482228896}{4166324251388991064948609718443166143} a^{11} + \frac{662656319945790050519798561140795559}{4166324251388991064948609718443166143} a^{10} - \frac{604665708560139418663584491572900507}{4166324251388991064948609718443166143} a^{9} - \frac{1169655561395226632684704893018648286}{4166324251388991064948609718443166143} a^{8} - \frac{577553212061915230998416762893239636}{4166324251388991064948609718443166143} a^{7} + \frac{949422922998497095682226396424424568}{4166324251388991064948609718443166143} a^{6} + \frac{167929076587199188248961366757380395}{378756750126271914995328156222106013} a^{5} - \frac{1437994461389627885599434064725158318}{4166324251388991064948609718443166143} a^{4} - \frac{697355821828296498969359390641813789}{4166324251388991064948609718443166143} a^{3} - \frac{1864662491909373799217463307088770350}{4166324251388991064948609718443166143} a^{2} + \frac{520865964717642579303596128300634600}{4166324251388991064948609718443166143} a + \frac{1734937658533183353617871578188015949}{4166324251388991064948609718443166143}$

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:  \( 3797541389.82 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T75:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24000
The 55 conjugacy class representatives for [F(5)^3]3=F(5)wr3 are not computed
Character table for [F(5)^3]3=F(5)wr3 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\)

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 R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $15$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]^{3}$
2.12.24.303$x^{12} + 32 x^{11} + 6 x^{10} + 32 x^{9} - 2 x^{8} + 16 x^{7} + 24 x^{6} - 8 x^{5} + 16 x^{3} + 8 x^{2} + 16 x - 24$$4$$3$$24$12T55$[2, 2, 3, 3]^{6}$
3Data not computed
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
73.4.3.3$x^{4} + 365$$4$$1$$3$$C_4$$[\ ]_{4}$
73.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
503Data not computed