Normalized defining polynomial
\( x^{16} - 3x^{14} + 8x^{12} + 3x^{10} - 9x^{8} + 3x^{6} + 8x^{4} - 3x^{2} + 1 \)
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: | \(350749278894882816\) \(\medspace = 2^{16}\cdot 3^{8}\cdot 13^{8}\) | 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: | \(12.49\) | 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: | $2\cdot 3^{1/2}13^{1/2}\approx 12.489995996796797$ | ||
Ramified primes: | \(2\), \(3\), \(13\) | 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: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{6}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{7}+\frac{1}{3}a^{3}$, $\frac{1}{21}a^{12}+\frac{1}{7}a^{6}+\frac{8}{21}$, $\frac{1}{21}a^{13}+\frac{1}{7}a^{7}+\frac{8}{21}a$, $\frac{1}{63}a^{14}+\frac{1}{63}a^{12}+\frac{1}{21}a^{8}-\frac{2}{7}a^{6}+\frac{1}{3}a^{4}+\frac{8}{63}a^{2}+\frac{29}{63}$, $\frac{1}{63}a^{15}+\frac{1}{63}a^{13}+\frac{1}{21}a^{9}-\frac{2}{7}a^{7}+\frac{1}{3}a^{5}+\frac{8}{63}a^{3}+\frac{29}{63}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
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: | \( \frac{5}{63} a^{15} + \frac{2}{63} a^{13} + \frac{40}{21} a^{9} + \frac{10}{7} a^{7} - \frac{5}{3} a^{5} + \frac{40}{63} a^{3} + \frac{100}{63} a \) (order $12$) | 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: | $\frac{23}{63}a^{15}-\frac{61}{63}a^{13}+\frac{8}{3}a^{11}+\frac{37}{21}a^{9}-\frac{40}{21}a^{7}+\frac{1}{3}a^{5}+\frac{163}{63}a^{3}+\frac{37}{63}a$, $\frac{5}{63}a^{14}-\frac{25}{63}a^{12}+a^{10}-\frac{16}{21}a^{8}-\frac{13}{7}a^{6}+\frac{2}{3}a^{4}+\frac{103}{63}a^{2}-\frac{95}{63}$, $\frac{25}{63}a^{15}-\frac{59}{63}a^{13}+\frac{7}{3}a^{11}+\frac{74}{21}a^{9}-\frac{80}{21}a^{7}-\frac{1}{3}a^{5}+\frac{221}{63}a^{3}-\frac{52}{63}a$, $\frac{5}{63}a^{15}+\frac{2}{63}a^{13}+\frac{40}{21}a^{9}+\frac{10}{7}a^{7}-\frac{5}{3}a^{5}+\frac{40}{63}a^{3}+\frac{100}{63}a-1$, $\frac{10}{63}a^{15}+\frac{5}{63}a^{14}-\frac{26}{63}a^{13}-\frac{25}{63}a^{12}+a^{11}+a^{10}+\frac{8}{7}a^{9}-\frac{16}{21}a^{8}-\frac{11}{7}a^{7}-\frac{13}{7}a^{6}-a^{5}+\frac{2}{3}a^{4}+\frac{143}{63}a^{3}+\frac{103}{63}a^{2}+\frac{44}{63}a-\frac{32}{63}$, $\frac{2}{7}a^{15}+\frac{8}{63}a^{14}-a^{13}-\frac{19}{63}a^{12}+\frac{8}{3}a^{11}+\frac{2}{3}a^{10}-\frac{1}{7}a^{9}+\frac{29}{21}a^{8}-\frac{10}{3}a^{7}-\frac{40}{21}a^{6}+2a^{5}-\frac{1}{3}a^{4}+\frac{41}{21}a^{3}+\frac{106}{63}a^{2}-a+\frac{16}{63}$, $\frac{1}{7}a^{15}-\frac{3}{7}a^{14}-\frac{1}{21}a^{13}+\frac{22}{21}a^{12}-\frac{8}{3}a^{10}+\frac{24}{7}a^{9}-\frac{23}{7}a^{8}-\frac{1}{7}a^{7}+\frac{73}{21}a^{6}-3a^{5}+a^{4}+\frac{8}{7}a^{3}-\frac{65}{21}a^{2}+\frac{34}{21}a-\frac{13}{21}$ | 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: | \( 771.781029552 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 771.781029552 \cdot 1}{12\cdot\sqrt{350749278894882816}}\cr\approx \mathstrut & 0.263787010940 \end{aligned}\]
Galois group
$C_2\times D_4$ (as 16T9):
A solvable group of order 16 |
The 10 conjugacy class representatives for $D_4\times C_2$ |
Character table for $D_4\times C_2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | 8.0.3504384.2, 8.0.65804544.2, 8.0.592240896.7, 8.4.592240896.2 |
Minimal sibling: | 8.0.3504384.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |