Normalized defining polynomial
\( x^{15} - 3 x^{14} + 2 x^{13} - x^{12} + 5 x^{11} - 22 x^{9} + 33 x^{8} - 22 x^{7} + 22 x^{6} - 22 x^{5} - 9 x^{4} + 38 x^{3} - 29 x^{2} + 9 x - 1 \)
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: | \(-388863829589238784\)\(\medspace = -\,2^{10}\cdot 11^{14}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $14.88$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{86} a^{13} - \frac{10}{43} a^{12} + \frac{1}{43} a^{11} + \frac{7}{43} a^{10} - \frac{5}{43} a^{9} - \frac{16}{43} a^{8} + \frac{9}{86} a^{7} + \frac{5}{43} a^{6} - \frac{27}{86} a^{5} - \frac{19}{43} a^{4} - \frac{16}{43} a^{2} + \frac{23}{86} a - \frac{16}{43}$, $\frac{1}{86} a^{14} - \frac{11}{86} a^{12} + \frac{11}{86} a^{11} + \frac{6}{43} a^{10} - \frac{17}{86} a^{9} - \frac{29}{86} a^{8} - \frac{25}{86} a^{7} - \frac{21}{43} a^{6} + \frac{12}{43} a^{5} - \frac{29}{86} a^{4} + \frac{11}{86} a^{3} - \frac{15}{86} a^{2} + \frac{41}{86} a + \frac{5}{86}$
Class group and class number
Trivial group, which has order $1$
Unit group
| 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: | \( 803.578266232599 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$C_5\times S_3$ (as 15T4):
| 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
| 3.1.484.1, \(\Q(\zeta_{11})^+\) |
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 | ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ | $15$ | ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ | R | $15$ | $15$ | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{5}$ | $15$ | ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ | $15$ | $15$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ | $15$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 2.10.10.7 | $x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
| $11$ | 11.15.14.1 | $x^{15} - 11$ | $15$ | $1$ | $14$ | $S_3 \times C_5$ | $[\ ]_{15}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (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) | $0$ | $1$ |
| 1.44.10t1.a.a | $1$ | $ 2^{2} \cdot 11 $ | 10.0.219503494144.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.44.10t1.a.b | $1$ | $ 2^{2} \cdot 11 $ | 10.0.219503494144.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.44.10t1.a.c | $1$ | $ 2^{2} \cdot 11 $ | 10.0.219503494144.1 | $C_{10}$ (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) | $0$ | $1$ |
| 1.44.10t1.a.d | $1$ | $ 2^{2} \cdot 11 $ | 10.0.219503494144.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| * | 2.484.3t2.b.a | $2$ | $ 2^{2} \cdot 11^{2}$ | 3.1.484.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.484.15t4.b.a | $2$ | $ 2^{2} \cdot 11^{2}$ | 15.5.388863829589238784.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
| * | 2.484.15t4.b.b | $2$ | $ 2^{2} \cdot 11^{2}$ | 15.5.388863829589238784.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
| * | 2.484.15t4.b.c | $2$ | $ 2^{2} \cdot 11^{2}$ | 15.5.388863829589238784.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
| * | 2.484.15t4.b.d | $2$ | $ 2^{2} \cdot 11^{2}$ | 15.5.388863829589238784.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |