↑

Properties

Label 15.15.4643879042...9152.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 3^{15}\cdot 11^{12}\cdot 23\cdot 43\cdot 3191^{2}$
Root discriminant $150.59$
Ramified primes $2, 3, 11, 23, 43, 3191$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T81

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-102112, -765840, -2297520, -3393512, -2268102, -63045, 701672, 219006, -72198, -35035, 3192, 2394, -52, -78, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 78*x^13 - 52*x^12 + 2394*x^11 + 3192*x^10 - 35035*x^9 - 72198*x^8 + 219006*x^7 + 701672*x^6 - 63045*x^5 - 2268102*x^4 - 3393512*x^3 - 2297520*x^2 - 765840*x - 102112)
 
gp: K = bnfinit(x^15 - 78*x^13 - 52*x^12 + 2394*x^11 + 3192*x^10 - 35035*x^9 - 72198*x^8 + 219006*x^7 + 701672*x^6 - 63045*x^5 - 2268102*x^4 - 3393512*x^3 - 2297520*x^2 - 765840*x - 102112, 1)
 

Normalized defining polynomial

\( x^{15} - 78 x^{13} - 52 x^{12} + 2394 x^{11} + 3192 x^{10} - 35035 x^{9} - 72198 x^{8} + 219006 x^{7} + 701672 x^{6} - 63045 x^{5} - 2268102 x^{4} - 3393512 x^{3} - 2297520 x^{2} - 765840 x - 102112 \)

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:  \(464387904272354847090911272369152=2^{10}\cdot 3^{15}\cdot 11^{12}\cdot 23\cdot 43\cdot 3191^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $150.59$
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:  $2, 3, 11, 23, 43, 3191$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{8} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{11} + \frac{5}{32} a^{10} - \frac{1}{8} a^{9} - \frac{15}{32} a^{8} - \frac{7}{16} a^{7} + \frac{5}{64} a^{6} + \frac{1}{4} a^{5} - \frac{15}{32} a^{4} - \frac{5}{16} a^{3} + \frac{11}{64} a^{2} - \frac{7}{16} a + \frac{3}{16}$, $\frac{1}{512} a^{13} - \frac{1}{128} a^{12} + \frac{7}{256} a^{11} + \frac{25}{128} a^{10} - \frac{7}{256} a^{9} + \frac{3}{16} a^{8} - \frac{131}{512} a^{7} + \frac{67}{256} a^{6} + \frac{65}{256} a^{5} + \frac{21}{64} a^{4} - \frac{77}{512} a^{3} - \frac{89}{256} a^{2} + \frac{1}{128} a + \frac{13}{64}$, $\frac{1}{4096} a^{14} + \frac{1}{2048} a^{13} - \frac{5}{2048} a^{12} + \frac{23}{512} a^{11} - \frac{731}{2048} a^{10} - \frac{125}{1024} a^{9} - \frac{67}{4096} a^{8} + \frac{93}{1024} a^{7} + \frac{211}{2048} a^{6} + \frac{237}{1024} a^{5} + \frac{419}{4096} a^{4} + \frac{11}{32} a^{3} + \frac{251}{512} a^{2} - \frac{11}{32} a - \frac{25}{256}$

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:  \( 340453427428 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T81:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 38880
The 63 conjugacy class representatives for [S(3)^5]5=S(3)wr5 are not computed
Character table for [S(3)^5]5=S(3)wr5 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\)

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
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 R R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R $15$ $15$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ R $15$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ R $15$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.9$x^{10} - 15 x^{8} + 38 x^{6} - 18 x^{4} + 25 x^{2} - 63$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
3Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
43Data not computed
3191Data not computed