Normalized defining polynomial
\( x^{16} - 7 x^{15} + 25 x^{14} - 55 x^{13} + 75 x^{12} - 46 x^{11} - 43 x^{10} + 140 x^{9} - 160 x^{8} + 80 x^{7} + 37 x^{6} - 109 x^{5} + 110 x^{4} - 70 x^{3} + 30 x^{2} - 8 x + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(6926910400390625\)\(\medspace = 5^{15}\cdot 61^{3}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $9.77$ | 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: | $5, 61$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Aut(K/\Q)|$: | $4$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1231} a^{15} - \frac{252}{1231} a^{14} + \frac{215}{1231} a^{13} + \frac{203}{1231} a^{12} - \frac{420}{1231} a^{11} - \frac{550}{1231} a^{10} + \frac{528}{1231} a^{9} + \frac{35}{1231} a^{8} - \frac{118}{1231} a^{7} - \frac{554}{1231} a^{6} + \frac{357}{1231} a^{5} - \frac{173}{1231} a^{4} - \frac{590}{1231} a^{3} + \frac{453}{1231} a^{2} - \frac{165}{1231} a - \frac{206}{1231}$
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: | \( \frac{6623}{1231} a^{15} - \frac{44076}{1231} a^{14} + \frac{149860}{1231} a^{13} - \frac{309995}{1231} a^{12} + \frac{383241}{1231} a^{11} - \frac{163844}{1231} a^{10} - \frac{343776}{1231} a^{9} + \frac{796834}{1231} a^{8} - \frac{764280}{1231} a^{7} + \frac{252824}{1231} a^{6} + \frac{327106}{1231} a^{5} - \frac{595522}{1231} a^{4} + \frac{512951}{1231} a^{3} - \frac{281627}{1231} a^{2} + \frac{100044}{1231} a - \frac{17624}{1231} \) (order $10$) | 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: | \( 65.2827160145 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
A solvable group of order 1024 |
The 88 conjugacy class representatives for t16n1192 are not computed |
Character table for t16n1192 is not computed |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.4765625.1 |
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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | $16$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | $16$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | $16$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
5 | Data not computed | ||||||
$61$ | 61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
61.4.3.1 | $x^{4} - 61$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |