Normalized defining polynomial
\( x^{15} - 6 x^{14} + 11 x^{13} - x^{12} - 10 x^{11} - 13 x^{10} + 18 x^{9} + 40 x^{8} - 30 x^{7} - 64 x^{6} + 42 x^{5} + 47 x^{4} - 27 x^{3} - 19 x^{2} + 9 x + 1 \)
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: | \(10544075126076649\)\(\medspace = 37^{2}\cdot 43^{2}\cdot 1609^{3}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $11.70$ | 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: | $37, 43, 1609$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{69781} a^{14} - \frac{26375}{69781} a^{13} - \frac{24841}{69781} a^{12} - \frac{1919}{69781} a^{11} + \frac{10876}{69781} a^{10} + \frac{10653}{69781} a^{9} + \frac{29367}{69781} a^{8} - \frac{18626}{69781} a^{7} + \frac{30286}{69781} a^{6} + \frac{31947}{69781} a^{5} - \frac{14169}{69781} a^{4} + \frac{14934}{69781} a^{3} - \frac{20490}{69781} a^{2} - \frac{13492}{69781} a + \frac{27019}{69781}$
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: | \( 73.2403388939 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
A non-solvable group of order 29160 |
The 108 conjugacy class representatives for [3^5]S(5)=3wrS(5) are not computed |
Character table for [3^5]S(5)=3wrS(5) is not computed |
Intermediate fields
5.1.1609.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 15 sibling: | 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 | $15$ | ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ | ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ | ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ | ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ | ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ | R | $15$ | R | ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
$37$ | 37.3.2.2 | $x^{3} + 74$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
37.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
37.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
37.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
$43$ | 43.3.2.2 | $x^{3} + 387$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
43.12.0.1 | $x^{12} - x + 33$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
1609 | Data not computed |