Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^13 - 5*x^12 + 8*x^11 + x^10 + 14*x^9 - 13*x^8 - 17*x^7 + 4*x^6 + x^5 + 22*x^4 - 2*x^3 - 8*x^2 - 3*x - 1)
gp: K = bnfinit(x^15 - 3*x^13 - 5*x^12 + 8*x^11 + x^10 + 14*x^9 - 13*x^8 - 17*x^7 + 4*x^6 + x^5 + 22*x^4 - 2*x^3 - 8*x^2 - 3*x - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, -8, -2, 22, 1, 4, -17, -13, 14, 1, 8, -5, -3, 0, 1]);
\( x^{15} - 3 x^{13} - 5 x^{12} + 8 x^{11} + x^{10} + 14 x^{9} - 13 x^{8} - 17 x^{7} + 4 x^{6} + x^{5} + 22 x^{4} - 2 x^{3} - 8 x^{2} - 3 x - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[5, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(-35351257235385344\)\(\medspace = -\,2^{10}\cdot 11^{13}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $12.68$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $2, 11$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $5$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2769251} a^{14} + \frac{1237085}{2769251} a^{13} - \frac{190661}{2769251} a^{12} - \frac{1217018}{2769251} a^{11} + \frac{1209397}{2769251} a^{10} + \frac{265482}{2769251} a^{9} - \frac{1060863}{2769251} a^{8} + \frac{806293}{2769251} a^{7} - \frac{772551}{2769251} a^{6} - \frac{1194966}{2769251} a^{5} - \frac{1253042}{2769251} a^{4} - \frac{753537}{2769251} a^{3} - \frac{1278776}{2769251} a^{2} - \frac{589461}{2769251} a - \frac{341613}{2769251}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
Class group and class number
Trivial group, which has order $1$
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
Unit group
sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
| Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| 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 | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 211.33204879 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$C_5\times S_3$ (as 15T4):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
| A solvable group of order 30 |
| The 15 conjugacy class representatives for $S_3 \times C_5$ |
| Character table for $S_3 \times C_5$ |
Intermediate fields
| 3.1.44.1, \(\Q(\zeta_{11})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $15$ | $15$ | ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ | ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ | $15$ | $15$ | ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ | $15$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
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]])
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];
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $11$ | 11.5.4.4 | $x^{5} - 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.10.9.7 | $x^{10} + 2673$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.11.5t1.a.a | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.11.5t1.a.b | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| 1.11.10t1.a.a | $1$ | $ 11 $ | \(\Q(\zeta_{11})\) | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.11.10t1.a.b | $1$ | $ 11 $ | \(\Q(\zeta_{11})\) | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.11.10t1.a.c | $1$ | $ 11 $ | \(\Q(\zeta_{11})\) | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| * | 1.11.5t1.a.c | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.11.5t1.a.d | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| 1.11.10t1.a.d | $1$ | $ 11 $ | \(\Q(\zeta_{11})\) | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| * | 2.44.3t2.b.a | $2$ | $ 2^{2} \cdot 11 $ | 3.1.44.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.484.15t4.a.a | $2$ | $ 2^{2} \cdot 11^{2}$ | 15.5.35351257235385344.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
| * | 2.484.15t4.a.b | $2$ | $ 2^{2} \cdot 11^{2}$ | 15.5.35351257235385344.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
| * | 2.484.15t4.a.c | $2$ | $ 2^{2} \cdot 11^{2}$ | 15.5.35351257235385344.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
| * | 2.484.15t4.a.d | $2$ | $ 2^{2} \cdot 11^{2}$ | 15.5.35351257235385344.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.