Normalized defining polynomial
\( x^{8} - 2x^{7} - 2x^{6} + 31x^{5} - 65x^{4} - 239x^{3} + 58x^{2} + 868x + 961 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(136744453125\) \(\medspace = 3^{6}\cdot 5^{7}\cdot 7^{4}\) | 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: | \(24.66\) | 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: | $3^{3/4}5^{7/8}7^{1/2}\approx 24.659752579382083$ | ||
Ramified primes: | \(3\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{4695477}a^{7}-\frac{4009}{53971}a^{6}+\frac{10188}{1565159}a^{5}+\frac{1672}{50489}a^{4}-\frac{1291091}{4695477}a^{3}-\frac{26677}{53971}a^{2}-\frac{77219}{161913}a+\frac{68816}{151467}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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: | $\frac{385}{161913}a^{7}-\frac{1607}{161913}a^{6}+\frac{1462}{161913}a^{5}+\frac{391}{5223}a^{4}-\frac{17032}{53971}a^{3}-\frac{2239}{161913}a^{2}+\frac{144532}{161913}a+\frac{8327}{5223}$, $\frac{9473}{4695477}a^{7}+\frac{1010}{161913}a^{6}-\frac{21643}{4695477}a^{5}+\frac{6419}{151467}a^{4}+\frac{1212542}{4695477}a^{3}-\frac{110968}{161913}a^{2}-\frac{350450}{161913}a+\frac{10163}{50489}$, $\frac{47065}{4695477}a^{7}-\frac{969}{53971}a^{6}+\frac{113539}{4695477}a^{5}+\frac{41965}{151467}a^{4}-\frac{2595217}{4695477}a^{3}-\frac{184838}{161913}a^{2}-\frac{120979}{161913}a+\frac{208135}{151467}$, $\frac{14806}{1565159}a^{7}-\frac{10328}{161913}a^{6}+\frac{199633}{1565159}a^{5}-\frac{2423}{50489}a^{4}-\frac{4949755}{4695477}a^{3}-\frac{5681}{53971}a^{2}+\frac{233034}{53971}a+\frac{1024319}{151467}$, $\frac{309698}{4695477}a^{7}-\frac{37223}{161913}a^{6}+\frac{1093199}{4695477}a^{5}+\frac{252061}{151467}a^{4}-\frac{31564286}{4695477}a^{3}-\frac{824047}{161913}a^{2}+\frac{551113}{53971}a+\frac{1990345}{50489}$ | 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: | \( 165.382367822 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 165.382367822 \cdot 2}{2\cdot\sqrt{136744453125}}\cr\approx \mathstrut & 0.282497146616 \end{aligned}\]
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 | R | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.6.3 | $x^{8} - 6 x^{4} + 18$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $[\ ]_{4}^{4}$ |
\(5\) | 5.8.7.1 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
\(7\) | 7.8.4.2 | $x^{8} + 245 x^{4} - 1372 x^{2} + 7203$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.15.4t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
1.5.4t1.a.a | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.5.4t1.a.b | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ | |
* | 1.15.4t1.a.b | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
* | 2.11025.8t7.a.a | $2$ | $ 3^{2} \cdot 5^{2} \cdot 7^{2}$ | 8.4.136744453125.1 | $C_8:C_2$ (as 8T7) | $0$ | $0$ |
* | 2.11025.8t7.a.b | $2$ | $ 3^{2} \cdot 5^{2} \cdot 7^{2}$ | 8.4.136744453125.1 | $C_8:C_2$ (as 8T7) | $0$ | $0$ |