Normalized defining polynomial
\( x^{8} - 2x^{7} - 32x^{6} + 16x^{5} + 325x^{4} + 121x^{3} - 1097x^{2} - 962x + 271 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(355444453125\) \(\medspace = 3^{6}\cdot 5^{7}\cdot 79^{2}\) | 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: | \(27.79\) | 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: | $3^{3/4}5^{7/8}79^{1/2}\approx 82.84250839896745$ | ||
Ramified primes: | \(3\), \(5\), \(79\) | 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: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{1713}a^{7}+\frac{269}{1713}a^{6}-\frac{221}{1713}a^{5}+\frac{80}{1713}a^{4}-\frac{88}{571}a^{3}-\frac{619}{1713}a^{2}+\frac{247}{571}a-\frac{572}{1713}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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: | \( -1 \) (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: | $\frac{7}{571}a^{7}-\frac{61}{1713}a^{6}-\frac{644}{1713}a^{5}+\frac{1109}{1713}a^{4}+\frac{5876}{1713}a^{3}-\frac{5005}{1713}a^{2}-\frac{16415}{1713}a+\frac{1121}{1713}$, $\frac{332}{1713}a^{7}-\frac{1481}{1713}a^{6}-\frac{7136}{1713}a^{5}+\frac{23134}{1713}a^{4}+\frac{55102}{1713}a^{3}-\frac{95305}{1713}a^{2}-\frac{51610}{571}a+\frac{44206}{1713}$, $\frac{163}{1713}a^{7}-\frac{691}{1713}a^{6}-\frac{3476}{1713}a^{5}+\frac{3395}{571}a^{4}+\frac{24917}{1713}a^{3}-\frac{39229}{1713}a^{2}-\frac{61366}{1713}a+\frac{16967}{1713}$, $\frac{376}{1713}a^{7}-\frac{1636}{1713}a^{6}-\frac{2765}{571}a^{5}+\frac{8504}{571}a^{4}+\frac{21728}{571}a^{3}-\frac{33995}{571}a^{2}-\frac{184465}{1713}a+\frac{24748}{1713}$, $\frac{2}{1713}a^{7}-\frac{11}{571}a^{6}+\frac{43}{571}a^{5}+\frac{160}{1713}a^{4}-\frac{1670}{1713}a^{3}+\frac{539}{571}a^{2}+\frac{1636}{571}a-\frac{7996}{1713}$, $\frac{332}{1713}a^{7}-\frac{1481}{1713}a^{6}-\frac{7136}{1713}a^{5}+\frac{23134}{1713}a^{4}+\frac{55102}{1713}a^{3}-\frac{95305}{1713}a^{2}-\frac{52181}{571}a+\frac{40780}{1713}$, $\frac{34}{1713}a^{7}+\frac{10}{1713}a^{6}-\frac{411}{571}a^{5}-\frac{1277}{1713}a^{4}+\frac{10438}{1713}a^{3}+\frac{4595}{571}a^{2}-\frac{23912}{1713}a-\frac{34865}{1713}$ | 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: | \( 1058.96238593 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 1058.96238593 \cdot 1}{2\cdot\sqrt{355444453125}}\cr\approx \mathstrut & 0.227355064762 \end{aligned}\]
Galois group
$\OD_{16}$ (as 8T7):
A solvable group of order 16 |
The 10 conjugacy class representatives for $C_8:C_2$ |
Character table for $C_8:C_2$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 16 |
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 | ${\href{/padicField/2.8.0.1}{8} }$ | R | R | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.6.3 | $x^{8} - 6 x^{4} + 18$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $[\ ]_{4}^{4}$ |
\(5\) | 5.8.7.1 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
\(79\) | 79.4.2.2 | $x^{4} - 6162 x^{2} + 18723$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
79.4.0.1 | $x^{4} + 2 x^{2} + 66 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |