Normalized defining polynomial
\( x^{16} + 34x^{14} + 425x^{12} + 2584x^{10} + 8670x^{8} + 16898x^{6} + 19040x^{4} + 11492x^{2} + 2873 \)
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: | \(31703006950533291639898112\) \(\medspace = 2^{16}\cdot 13^{2}\cdot 17^{15}\) | 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: | \(39.25\) | 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: | $2^{3/2}13^{1/2}17^{15/16}\approx 145.23207201172457$ | ||
Ramified primes: | \(2\), \(13\), \(17\) | 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: | \(\Q(\sqrt{17}) \) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{3}+\frac{1}{4}a$, $\frac{1}{52}a^{10}-\frac{25}{52}a^{6}+\frac{1}{26}a^{4}-\frac{25}{52}a^{2}+\frac{1}{4}$, $\frac{1}{52}a^{11}-\frac{25}{52}a^{7}+\frac{1}{26}a^{5}-\frac{25}{52}a^{3}+\frac{1}{4}a$, $\frac{1}{52}a^{12}+\frac{1}{52}a^{8}-\frac{6}{13}a^{6}-\frac{25}{52}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{52}a^{13}+\frac{1}{52}a^{9}-\frac{6}{13}a^{7}-\frac{25}{52}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{52}a^{14}+\frac{1}{26}a^{8}-\frac{1}{2}a^{6}+\frac{11}{52}a^{4}-\frac{1}{52}a^{2}+\frac{1}{4}$, $\frac{1}{52}a^{15}+\frac{1}{26}a^{9}-\frac{1}{2}a^{7}+\frac{11}{52}a^{5}-\frac{1}{52}a^{3}+\frac{1}{4}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{50}$, which has order $200$ (assuming GRH)
Unit group
Rank: | $7$ | 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)
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Fundamental units: | $\frac{11}{52}a^{14}+\frac{361}{52}a^{12}+\frac{1063}{13}a^{10}+\frac{11755}{26}a^{8}+\frac{68739}{52}a^{6}+\frac{27447}{13}a^{4}+\frac{22650}{13}a^{2}+580$, $\frac{2}{13}a^{14}+\frac{261}{52}a^{12}+\frac{3041}{52}a^{10}+\frac{4122}{13}a^{8}+\frac{11671}{13}a^{6}+\frac{71145}{52}a^{4}+\frac{27679}{26}a^{2}+\frac{667}{2}$, $\frac{11}{26}a^{14}+\frac{737}{52}a^{12}+\frac{8963}{52}a^{10}+\frac{51793}{52}a^{8}+\frac{158741}{52}a^{6}+\frac{264111}{52}a^{4}+\frac{56052}{13}a^{2}+\frac{5819}{4}$, $\frac{41}{52}a^{14}+\frac{1345}{52}a^{12}+\frac{15819}{52}a^{10}+\frac{21771}{13}a^{8}+\frac{125949}{26}a^{6}+\frac{98559}{13}a^{4}+\frac{315259}{52}a^{2}+\frac{7747}{4}$, $\frac{3}{26}a^{14}+\frac{211}{52}a^{12}+\frac{2757}{52}a^{10}+\frac{4375}{13}a^{8}+\frac{29663}{26}a^{6}+\frac{108829}{52}a^{4}+\frac{50483}{26}a^{2}+\frac{1417}{2}$, $\frac{19}{26}a^{14}+\frac{623}{26}a^{12}+\frac{14639}{52}a^{10}+\frac{20097}{13}a^{8}+\frac{231267}{52}a^{6}+\frac{179063}{26}a^{4}+\frac{281257}{52}a^{2}+\frac{6735}{4}$, $\frac{41}{52}a^{14}+\frac{1345}{52}a^{12}+\frac{15819}{52}a^{10}+\frac{21771}{13}a^{8}+\frac{125949}{26}a^{6}+\frac{98559}{13}a^{4}+\frac{315259}{52}a^{2}+\frac{7743}{4}$ (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: | \( 3640.01221338 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3640.01221338 \cdot 200}{2\cdot\sqrt{31703006950533291639898112}}\cr\approx \mathstrut & 0.157033228560 \end{aligned}\] (assuming GRH)
Galois group
$C_2^4.C_8$ (as 16T306):
A solvable group of order 128 |
The 26 conjugacy class representatives for $C_2^4.C_8$ |
Character table for $C_2^4.C_8$ |
Intermediate fields
\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\) |
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 siblings: | 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 | $16$ | $16$ | $16$ | $16$ | R | R | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.5 | $x^{8} + 8 x^{7} + 56 x^{6} + 184 x^{5} + 576 x^{4} + 960 x^{3} + 1632 x^{2} + 1120 x + 304$ | $2$ | $4$ | $8$ | $C_8:C_2$ | $[2, 2]^{4}$ |
2.8.8.5 | $x^{8} + 8 x^{7} + 56 x^{6} + 184 x^{5} + 576 x^{4} + 960 x^{3} + 1632 x^{2} + 1120 x + 304$ | $2$ | $4$ | $8$ | $C_8:C_2$ | $[2, 2]^{4}$ | |
\(13\) | 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(17\) | 17.16.15.1 | $x^{16} + 272$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |