Normalized defining polynomial
\( x^{15} - x^{14} - x^{13} + 2 x^{12} - 4 x^{11} - x^{10} + 3 x^{9} + 3 x^{8} + 12 x^{7} - 2 x^{6} - 14 x^{5} + 7 x^{4} + 11 x^{3} - 6 x^{2} + 1 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
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Signature: | $[1, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(-6401597515801839\)\(\medspace = -\,3^{9}\cdot 13^{3}\cdot 23^{6}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $11.32$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
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Ramified primes: | $3, 13, 23$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Aut(K/\Q)|$: | $1$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{23} a^{13} - \frac{7}{23} a^{11} - \frac{5}{23} a^{10} + \frac{10}{23} a^{9} - \frac{7}{23} a^{8} + \frac{5}{23} a^{7} + \frac{4}{23} a^{6} + \frac{9}{23} a^{5} + \frac{6}{23} a^{4} + \frac{7}{23} a^{3} + \frac{1}{23} a^{2} - \frac{7}{23} a + \frac{4}{23}$, $\frac{1}{1917533} a^{14} + \frac{29448}{1917533} a^{13} - \frac{761330}{1917533} a^{12} - \frac{611332}{1917533} a^{11} - \frac{232445}{1917533} a^{10} + \frac{820230}{1917533} a^{9} + \frac{707153}{1917533} a^{8} - \frac{293390}{1917533} a^{7} - \frac{138626}{1917533} a^{6} + \frac{781020}{1917533} a^{5} - \frac{300256}{1917533} a^{4} - \frac{327532}{1917533} a^{3} + \frac{534843}{1917533} a^{2} + \frac{642407}{1917533} a + \frac{946988}{1917533}$
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
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Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
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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)];
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Regulator: | \( 46.8775403574 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
$S_5 \times S_3$ (as 15T29):
A non-solvable group of order 720 |
The 21 conjugacy class representatives for $S_5 \times S_3$ |
Character table for $S_5 \times S_3$ is not computed |
Intermediate fields
3.1.23.1, 5.1.8073.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 18 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 36 siblings: | data not computed |
Degree 45 sibling: | 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 | $15$ | R | ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ | R | ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ | ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
$3$ | 3.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
3.9.9.1 | $x^{9} + 54 x^{5} + 27 x^{3} + 189$ | $3$ | $3$ | $9$ | $S_3\times C_3$ | $[3/2]_{2}^{3}$ | |
$13$ | 13.3.0.1 | $x^{3} - 2 x + 6$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
13.3.0.1 | $x^{3} - 2 x + 6$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
13.3.0.1 | $x^{3} - 2 x + 6$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
13.6.3.1 | $x^{6} - 52 x^{4} + 676 x^{2} - 79092$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
$23$ | 23.2.1.1 | $x^{2} - 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
23.3.0.1 | $x^{3} - x + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
23.6.3.2 | $x^{6} - 529 x^{2} + 48668$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |