Normalized defining polynomial
\( x^{15} - x^{14} - x^{13} + x^{12} - 7 x^{11} - 2 x^{10} + 18 x^{9} + 12 x^{8} - 17 x^{7} + 7 x^{6} - 11 x^{5} - 21 x^{4} + 22 x^{3} - 11 x^{2} - 2 x - 1 \)
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: | \(21643467887730481\)\(\medspace = 13^{2}\cdot 47^{6}\cdot 109^{2}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $12.28$ | 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: | $13, 47, 109$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Aut(K/\Q)|$: | $3$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4823316529} a^{14} - \frac{240451914}{4823316529} a^{13} - \frac{1467210693}{4823316529} a^{12} + \frac{2044776038}{4823316529} a^{11} + \frac{2380646120}{4823316529} a^{10} - \frac{2290615831}{4823316529} a^{9} - \frac{1561717558}{4823316529} a^{8} + \frac{2080171999}{4823316529} a^{7} - \frac{2386498101}{4823316529} a^{6} + \frac{1715950732}{4823316529} a^{5} + \frac{1834453739}{4823316529} a^{4} + \frac{64937028}{4823316529} a^{3} + \frac{593205418}{4823316529} a^{2} + \frac{970975456}{4823316529} a + \frac{820723294}{4823316529}$
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: | \( 116.633322592 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
A solvable group of order 2430 |
The 72 conjugacy class representatives for [3^5]D(5)=3wrD(5) are not computed |
Character table for [3^5]D(5)=3wrD(5) is not computed |
Intermediate fields
5.1.2209.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 15 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 45 siblings: | 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 | $15$ | ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ | $15$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ | R | $15$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ | $15$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ | R | $15$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
$13$ | 13.3.2.2 | $x^{3} - 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
13.6.0.1 | $x^{6} + x^{2} - 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
13.6.0.1 | $x^{6} + x^{2} - 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
$47$ | 47.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
47.6.3.2 | $x^{6} - 2209 x^{2} + 207646$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
47.6.3.2 | $x^{6} - 2209 x^{2} + 207646$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
$109$ | 109.3.2.2 | $x^{3} + 654$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
109.6.0.1 | $x^{6} - x + 11$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
109.6.0.1 | $x^{6} - x + 11$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |