Normalized defining polynomial
\( x^{15} - 4 x^{14} + 4 x^{13} + 4 x^{12} - 5 x^{11} - 13 x^{10} + 20 x^{9} + 4 x^{8} - 15 x^{7} - 13 x^{6} + 27 x^{5} - 4 x^{4} - 8 x^{3} - 2 x^{2} + 6 x - 1 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
Signature: | $[1, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
Discriminant: | \(-44543599279432079\)\(\medspace = -\,239^{7}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
Root discriminant: | $12.88$ | 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: | $239$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
$|\Aut(K/\Q)|$: | $1$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{34333} a^{14} - \frac{6031}{34333} a^{13} - \frac{9806}{34333} a^{12} + \frac{13673}{34333} a^{11} - \frac{7976}{34333} a^{10} + \frac{5139}{34333} a^{9} - \frac{4367}{34333} a^{8} - \frac{13498}{34333} a^{7} - \frac{16779}{34333} a^{6} + \frac{16335}{34333} a^{5} + \frac{16026}{34333} a^{4} - \frac{9977}{34333} a^{3} + \frac{752}{1807} a^{2} - \frac{6614}{34333} a + \frac{1971}{34333}$
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | \( 124.657592501 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
A solvable group of order 30 |
The 9 conjugacy class representatives for $D_{15}$ |
Character table for $D_{15}$ |
Intermediate fields
3.1.239.1, 5.1.57121.1 |
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 | $15$ | ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ | $15$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | $15$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{5}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | $15$ | $15$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
239 | Data not computed |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.239.2t1.a.a | $1$ | $ 239 $ | \(\Q(\sqrt{-239}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.239.3t2.a.a | $2$ | $ 239 $ | 3.1.239.1 | $S_3$ (as 3T2) | $1$ | $0$ |
* | 2.239.5t2.a.a | $2$ | $ 239 $ | 5.1.57121.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.239.5t2.a.b | $2$ | $ 239 $ | 5.1.57121.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.239.15t2.a.c | $2$ | $ 239 $ | 15.1.44543599279432079.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.239.15t2.a.a | $2$ | $ 239 $ | 15.1.44543599279432079.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.239.15t2.a.b | $2$ | $ 239 $ | 15.1.44543599279432079.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.239.15t2.a.d | $2$ | $ 239 $ | 15.1.44543599279432079.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |