Normalized defining polynomial
\( x^{16} - 5x^{14} + 23x^{10} - 11x^{8} - 28x^{6} + 30x^{4} - 10x^{2} + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[12, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(307463056776831238144\) \(\medspace = 2^{16}\cdot 811^{2}\cdot 84457^{2}\) | 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: | \(19.08\) | 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))
| |
Galois root discriminant: | not computed | ||
Ramified primes: | \(2\), \(811\), \(84457\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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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 $1$ (assuming GRH)
Unit group
Rank: | $13$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $a$, $81a^{15}-360a^{13}-200a^{11}+1752a^{9}+82a^{7}-2223a^{5}+1196a^{3}-144a$, $96a^{15}-428a^{13}-232a^{11}+2083a^{9}+73a^{7}-2652a^{5}+1443a^{3}-175a$, $114a^{15}-509a^{13}-272a^{11}+2475a^{9}+69a^{7}-3148a^{5}+1738a^{3}-219a$, $98a^{15}-437a^{13}-236a^{11}+2125a^{9}+70a^{7}-2700a^{5}+1483a^{3}-185a$, $17a^{14}-76a^{12}-40a^{10}+369a^{8}+7a^{6}-468a^{4}+265a^{2}-35$, $149a^{15}+16a^{14}-664a^{13}-72a^{12}-361a^{11}-36a^{10}+3231a^{9}+350a^{8}+117a^{7}-a^{6}-4109a^{5}-448a^{4}+2238a^{3}+255a^{2}-273a-33$, $81a^{15}-360a^{13}-200a^{11}+1752a^{9}+82a^{7}-2223a^{5}+1196a^{3}-144a-1$, $a^{15}+17a^{14}-5a^{13}-76a^{12}-40a^{10}+23a^{9}+369a^{8}-11a^{7}+7a^{6}-28a^{5}-468a^{4}+30a^{3}+265a^{2}-10a-35$, $33a^{15}-96a^{14}-149a^{13}+428a^{12}-72a^{11}+232a^{10}+723a^{9}-2083a^{8}-13a^{7}-73a^{6}-925a^{5}+2652a^{4}+542a^{3}-1442a^{2}-75a+174$, $51a^{15}-96a^{14}-227a^{13}+428a^{12}-125a^{11}+232a^{10}+1106a^{9}-2083a^{8}+47a^{7}-73a^{6}-1409a^{5}+2652a^{4}+755a^{3}-1443a^{2}-88a+175$, $17a^{15}+61a^{14}-77a^{13}-273a^{12}-36a^{11}-143a^{10}+373a^{9}+1327a^{8}-12a^{7}+25a^{6}-477a^{5}-1691a^{4}+287a^{3}+942a^{2}-41a-119$, $a^{15}+45a^{14}-5a^{13}-201a^{12}-107a^{10}+23a^{9}+977a^{8}-11a^{7}+26a^{6}-28a^{5}-1243a^{4}+30a^{3}+687a^{2}-9a-86$ (assuming GRH) | 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: | \( 32162.7474809 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{2}\cdot 32162.7474809 \cdot 1}{2\cdot\sqrt{307463056776831238144}}\cr\approx \mathstrut & 0.148301778632 \end{aligned}\] (assuming GRH)
Galois group
$C_2^7.S_8$ (as 16T1945):
A non-solvable group of order 5160960 |
The 100 conjugacy class representatives for $C_2^7.S_8$ |
Character table for $C_2^7.S_8$ |
Intermediate fields
8.6.68494627.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 16 sibling: | data not computed |
Degree 32 sibling: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $16$ | $2$ | $8$ | $16$ | |||
\(811\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(84457\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |