Normalized defining polynomial
\( x^{16} + 17x^{12} - 32x^{8} + 17x^{4} + 1 \)
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: | \(29960650073923649536\) \(\medspace = 2^{32}\cdot 17^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.49\) | 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: | $2^{2}17^{1/2}\approx 16.492422502470642$ | ||
Ramified primes: | \(2\), \(17\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5}a^{8}+\frac{1}{5}a^{4}+\frac{1}{5}$, $\frac{1}{5}a^{9}+\frac{1}{5}a^{5}+\frac{1}{5}a$, $\frac{1}{5}a^{10}+\frac{1}{5}a^{6}+\frac{1}{5}a^{2}$, $\frac{1}{5}a^{11}+\frac{1}{5}a^{7}+\frac{1}{5}a^{3}$, $\frac{1}{80}a^{12}-\frac{1}{10}a^{10}-\frac{1}{10}a^{6}-\frac{1}{10}a^{2}-\frac{31}{80}$, $\frac{1}{80}a^{13}-\frac{1}{10}a^{11}-\frac{1}{10}a^{7}-\frac{1}{10}a^{3}-\frac{31}{80}a$, $\frac{1}{80}a^{14}-\frac{1}{10}a^{8}-\frac{1}{10}a^{4}-\frac{31}{80}a^{2}-\frac{1}{10}$, $\frac{1}{80}a^{15}-\frac{1}{10}a^{9}-\frac{1}{10}a^{5}-\frac{31}{80}a^{3}-\frac{1}{10}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{13}{40} a^{15} + \frac{27}{5} a^{11} - \frac{63}{5} a^{7} + \frac{333}{40} a^{3} \) (order $8$) | 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: | $\frac{3}{16}a^{15}+\frac{1}{40}a^{13}+\frac{16}{5}a^{11}+\frac{1}{2}a^{9}-\frac{29}{5}a^{7}+\frac{1}{2}a^{5}+\frac{191}{80}a^{3}-\frac{51}{40}a$, $\frac{3}{16}a^{14}+\frac{1}{40}a^{12}+\frac{16}{5}a^{10}+\frac{1}{2}a^{8}-\frac{29}{5}a^{6}+\frac{1}{2}a^{4}+\frac{191}{80}a^{2}-\frac{11}{40}$, $\frac{13}{40}a^{15}+\frac{1}{40}a^{14}+\frac{9}{80}a^{12}+\frac{27}{5}a^{11}+\frac{1}{2}a^{10}+2a^{8}-\frac{63}{5}a^{7}+\frac{1}{2}a^{6}-2a^{4}+\frac{333}{40}a^{3}-\frac{51}{40}a^{2}+\frac{41}{80}$, $\frac{1}{40}a^{14}-\frac{3}{40}a^{13}-\frac{9}{80}a^{12}+\frac{1}{2}a^{10}-\frac{6}{5}a^{9}-2a^{8}+\frac{1}{2}a^{6}+\frac{19}{5}a^{5}+2a^{4}-\frac{51}{40}a^{2}-\frac{15}{8}a-\frac{41}{80}$, $\frac{13}{40}a^{15}+\frac{3}{10}a^{14}+\frac{3}{40}a^{13}+\frac{27}{5}a^{11}+\frac{26}{5}a^{10}+\frac{6}{5}a^{9}-\frac{63}{5}a^{7}-\frac{39}{5}a^{6}-\frac{19}{5}a^{5}+\frac{333}{40}a^{3}+\frac{39}{10}a^{2}+\frac{15}{8}a$, $\frac{1}{8}a^{14}+\frac{11}{5}a^{10}-\frac{14}{5}a^{6}-\frac{67}{40}a^{2}$, $\frac{37}{80}a^{15}-\frac{7}{16}a^{14}+\frac{13}{80}a^{13}-\frac{1}{10}a^{12}+\frac{79}{10}a^{11}-\frac{37}{5}a^{10}+\frac{27}{10}a^{9}-\frac{17}{10}a^{8}-\frac{141}{10}a^{7}+\frac{73}{5}a^{6}-\frac{63}{10}a^{5}+\frac{33}{10}a^{4}+\frac{121}{16}a^{3}-\frac{707}{80}a^{2}+\frac{293}{80}a-\frac{3}{5}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 4179.28334772 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4179.28334772 \cdot 2}{8\cdot\sqrt{29960650073923649536}}\cr\approx \mathstrut & 0.463665898652 \end{aligned}\]
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: | 8.0.18939904.4, 8.0.5473632256.6, 8.4.1368408064.3, 8.0.1368408064.4 |
Minimal sibling: | 8.0.18939904.4 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ |
2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
\(17\) | 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |