Normalized defining polynomial
\( x^{16} - 2 x^{15} - 2 x^{14} + 9 x^{13} - 3 x^{12} - 15 x^{11} + 17 x^{10} + 8 x^{9} - 25 x^{8} + \cdots + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(3692461181640625\) \(\medspace = 5^{12}\cdot 3889^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(9.40\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
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Galois root discriminant: | $5^{3/4}3889^{1/2}\approx 208.51940394717084$ | ||
Ramified primes: | \(5\), \(3889\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -a^{15} + 3 a^{14} + a^{13} - 12 a^{12} + 9 a^{11} + 17 a^{10} - 28 a^{9} - a^{8} + 33 a^{7} - 20 a^{6} - 15 a^{5} + 21 a^{4} - 2 a^{3} - 8 a^{2} + 3 a + 1 \) (order $10$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $a^{15}-3a^{14}-a^{13}+11a^{12}-7a^{11}-16a^{10}+21a^{9}+5a^{8}-26a^{7}+9a^{6}+16a^{5}-14a^{4}-4a^{3}+7a^{2}-2$, $a^{13}-2a^{12}-a^{11}+7a^{10}-4a^{9}-8a^{8}+12a^{7}+a^{6}-11a^{5}+5a^{4}+5a^{3}-5a^{2}-a+1$, $2a^{15}-2a^{14}-6a^{13}+11a^{12}+6a^{11}-20a^{10}+7a^{9}+18a^{8}-16a^{7}-4a^{6}+13a^{5}-2a^{4}-3a^{3}+a^{2}+a+1$, $a^{15}-a^{14}-3a^{13}+6a^{12}+3a^{11}-12a^{10}+5a^{9}+12a^{8}-11a^{7}-2a^{6}+9a^{5}-3a^{4}-a^{3}+a^{2}+1$, $2a^{15}-4a^{14}-4a^{13}+17a^{12}-5a^{11}-27a^{10}+28a^{9}+14a^{8}-39a^{7}+9a^{6}+26a^{5}-18a^{4}-8a^{3}+11a^{2}+a-3$, $a^{14}-2a^{13}-2a^{12}+9a^{11}-3a^{10}-14a^{9}+16a^{8}+6a^{7}-21a^{6}+9a^{5}+12a^{4}-11a^{3}+5a-1$, $2a^{15}-4a^{14}-3a^{13}+16a^{12}-7a^{11}-22a^{10}+28a^{9}+7a^{8}-31a^{7}+13a^{6}+16a^{5}-14a^{4}-a^{3}+5a^{2}-1$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 40.8735633138 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 40.8735633138 \cdot 1}{10\cdot\sqrt{3692461181640625}}\cr\approx \mathstrut & 0.163389249678 \end{aligned}\]
Galois group
$S_4^2:C_4$ (as 16T1497):
A solvable group of order 2304 |
The 40 conjugacy class representatives for $S_4^2:C_4$ |
Character table for $S_4^2:C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.4.60765625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | 12.0.1181587578125.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.4.0.1}{4} }$ | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |
\(3889\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |