Normalized defining polynomial
\( x^{16} - 3x^{12} + 64x^{8} + 37x^{4} + 1 \)
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)
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2120}a^{12}-\frac{1}{2}a^{10}+\frac{101}{265}a^{8}-\frac{1}{2}a^{6}+\frac{34}{265}a^{4}-\frac{1}{2}a^{2}+\frac{149}{2120}$, $\frac{1}{2120}a^{13}-\frac{1}{2}a^{11}+\frac{101}{265}a^{9}-\frac{1}{2}a^{7}+\frac{34}{265}a^{5}-\frac{1}{2}a^{3}+\frac{149}{2120}a$, $\frac{1}{2120}a^{14}+\frac{101}{265}a^{10}-\frac{1}{2}a^{8}+\frac{34}{265}a^{6}-\frac{1}{2}a^{4}+\frac{149}{2120}a^{2}-\frac{1}{2}$, $\frac{1}{2120}a^{15}+\frac{101}{265}a^{11}-\frac{1}{2}a^{9}+\frac{34}{265}a^{7}-\frac{1}{2}a^{5}+\frac{149}{2120}a^{3}-\frac{1}{2}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | \( -\frac{23}{106} a^{14} + \frac{36}{53} a^{10} - \frac{743}{53} a^{6} - \frac{671}{106} a^{2} \) (order $4$) | 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: | $\frac{183}{2120}a^{14}-\frac{49}{1060}a^{12}-\frac{67}{265}a^{10}+\frac{79}{530}a^{8}+\frac{1452}{265}a^{6}-\frac{1629}{530}a^{4}+\frac{8187}{2120}a^{2}-\frac{411}{1060}$, $\frac{1}{530}a^{14}-\frac{81}{2120}a^{12}+\frac{13}{530}a^{10}+\frac{34}{265}a^{8}+\frac{7}{530}a^{6}-\frac{634}{265}a^{4}+\frac{472}{265}a^{2}-\frac{1469}{2120}$, $\frac{183}{2120}a^{15}+\frac{49}{1060}a^{13}-\frac{67}{265}a^{11}-\frac{79}{530}a^{9}+\frac{1452}{265}a^{7}+\frac{1629}{530}a^{5}+\frac{8187}{2120}a^{3}+\frac{1471}{1060}a$, $\frac{647}{2120}a^{15}+\frac{281}{2120}a^{14}-\frac{281}{2120}a^{13}+\frac{179}{2120}a^{12}-\frac{481}{530}a^{11}-\frac{213}{530}a^{10}+\frac{213}{530}a^{9}-\frac{147}{530}a^{8}+\frac{10341}{530}a^{7}+\frac{4533}{530}a^{6}-\frac{4533}{530}a^{5}+\frac{2897}{530}a^{4}+\frac{25383}{2120}a^{3}+\frac{9009}{2120}a^{2}-\frac{11129}{2120}a+\frac{4411}{2120}$, $\frac{1}{530}a^{15}+\frac{281}{2120}a^{14}-\frac{183}{2120}a^{13}-\frac{17}{2120}a^{12}+\frac{13}{530}a^{11}-\frac{213}{530}a^{10}+\frac{67}{265}a^{9}+\frac{11}{530}a^{8}+\frac{7}{530}a^{7}+\frac{4533}{530}a^{6}-\frac{1452}{265}a^{5}-\frac{361}{530}a^{4}+\frac{472}{265}a^{3}+\frac{9009}{2120}a^{2}-\frac{6067}{2120}a+\frac{647}{2120}$, $\frac{411}{530}a^{15}-\frac{187}{2120}a^{14}-\frac{77}{2120}a^{12}-\frac{641}{265}a^{11}+\frac{121}{530}a^{10}+\frac{81}{530}a^{8}+\frac{13231}{265}a^{7}-\frac{2911}{530}a^{6}-\frac{1261}{530}a^{4}+\frac{11949}{530}a^{3}-\frac{11963}{2120}a^{2}+\frac{2307}{2120}$, $\frac{23}{106}a^{15}+\frac{183}{2120}a^{14}-\frac{51}{1060}a^{13}+\frac{49}{1060}a^{12}-\frac{36}{53}a^{11}-\frac{67}{265}a^{10}+\frac{33}{265}a^{9}-\frac{79}{530}a^{8}+\frac{743}{53}a^{7}+\frac{1452}{265}a^{6}-\frac{818}{265}a^{5}+\frac{1629}{530}a^{4}+\frac{671}{106}a^{3}+\frac{6067}{2120}a^{2}-\frac{2299}{1060}a+\frac{1471}{1060}$ | 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: | \( 3446.93143062 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3446.93143062 \cdot 1}{4\cdot\sqrt{29960650073923649536}}\cr\approx \mathstrut & 0.382415937469 \end{aligned}\]
Galois group
A solvable group of order 16 |
The 7 conjugacy class representatives for $D_{8}$ |
Character table for $D_{8}$ |
Intermediate fields
\(\Q(\sqrt{-17}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{17})\), 4.2.1156.1 x2, 4.0.272.1 x2, 8.0.21381376.2, 8.2.1368408064.1 x4, 8.0.321978368.4 x4 |
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 | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.1.0.1}{1} }^{16}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.16.5 | $x^{8} + 4 x^{7} + 14 x^{6} + 36 x^{5} + 73 x^{4} + 88 x^{3} + 48 x^{2} + 56 x + 61$ | $4$ | $2$ | $16$ | $D_4$ | $[2, 3]^{2}$ |
2.8.16.5 | $x^{8} + 4 x^{7} + 14 x^{6} + 36 x^{5} + 73 x^{4} + 88 x^{3} + 48 x^{2} + 56 x + 61$ | $4$ | $2$ | $16$ | $D_4$ | $[2, 3]^{2}$ | |
\(17\) | 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |