Normalized defining polynomial
\( x^{15} - 2 x^{14} + 6 x^{13} - 14 x^{12} + 23 x^{11} - 35 x^{10} + 44 x^{9} - 41 x^{8} + 25 x^{7} - 4 x^{6} - 20 x^{5} + 34 x^{4} - 30 x^{3} + 17 x^{2} - 6 x + 1 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
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Signature: | $[3, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(16680932154277329\)\(\medspace = 3\cdot 47\cdot 103\cdot 11057\cdot 103878739\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $12.06$ | 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: | $3, 47, 103, 11057, 103878739$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{229} a^{14} - \frac{80}{229} a^{13} + \frac{63}{229} a^{12} + \frac{110}{229} a^{11} - \frac{84}{229} a^{10} + \frac{105}{229} a^{9} + \frac{98}{229} a^{8} + \frac{101}{229} a^{7} - \frac{67}{229} a^{6} - \frac{45}{229} a^{5} + \frac{55}{229} a^{4} + \frac{95}{229} a^{3} - \frac{112}{229} a^{2} + \frac{51}{229} a - \frac{91}{229}$
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | \( 124.641970008 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
A non-solvable group of order 1307674368000 |
The 176 conjugacy class representatives for S15 are not computed |
Character table for S15 is not computed |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 30 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 | $15$ | ${\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ | ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | $15$ | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ | ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ | ${\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.11.0.1}{11} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.11.0.1}{11} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.13.0.1}{13} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
$3$ | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
3.9.0.1 | $x^{9} - x^{3} + x^{2} + 1$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
$47$ | 47.2.1.2 | $x^{2} + 94$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
47.13.0.1 | $x^{13} - x + 32$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
$103$ | 103.2.1.2 | $x^{2} + 206$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
103.5.0.1 | $x^{5} - x + 18$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
103.8.0.1 | $x^{8} - x + 6$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
11057 | Data not computed | ||||||
103878739 | Data not computed |