Normalized defining polynomial
\( x^{8} - 2x^{7} - 7x^{6} - 4x^{5} + 70x^{4} + 41x^{3} - 287x^{2} - 62x + 401 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1340703125\) \(\medspace = 5^{7}\cdot 131^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.83\) | 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))
| |
Galois root discriminant: | $5^{7/8}131^{1/2}\approx 46.79876599652796$ | ||
Ramified primes: | \(5\), \(131\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{141949}a^{7}+\frac{21067}{141949}a^{6}-\frac{13907}{141949}a^{5}-\frac{23851}{141949}a^{4}-\frac{17189}{141949}a^{3}-\frac{43101}{141949}a^{2}-\frac{47503}{141949}a+\frac{41630}{141949}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{10518}{141949} a^{7} + \frac{317}{141949} a^{6} - \frac{66356}{141949} a^{5} - \frac{182884}{141949} a^{4} + \frac{333022}{141949} a^{3} + \frac{1042431}{141949} a^{2} - \frac{685819}{141949} a - \frac{1751713}{141949} \) (order $10$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{100}{7471}a^{7}-\frac{122}{7471}a^{6}-\frac{1094}{7471}a^{5}-\frac{1851}{7471}a^{4}+\frac{6901}{7471}a^{3}+\frac{15609}{7471}a^{2}-\frac{13686}{7471}a-\frac{43173}{7471}$, $\frac{714}{141949}a^{7}-\frac{4756}{141949}a^{6}+\frac{6832}{141949}a^{5}+\frac{4266}{141949}a^{4}+\frac{76617}{141949}a^{3}-\frac{255079}{141949}a^{2}+\frac{8669}{141949}a+\frac{340377}{141949}$, $\frac{13592}{141949}a^{7}+\frac{31531}{141949}a^{6}-\frac{89825}{141949}a^{5}-\frac{397123}{141949}a^{4}-\frac{126783}{141949}a^{3}+\frac{1982017}{141949}a^{2}+\frac{1058868}{141949}a-\frac{3522479}{141949}$ | 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: | \( 50.8310364801 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 50.8310364801 \cdot 1}{10\cdot\sqrt{1340703125}}\cr\approx \mathstrut & 0.216362568265 \end{aligned}\]
Galois group
$\OD_{16}:C_2$ (as 8T16):
A solvable group of order 32 |
The 11 conjugacy class representatives for $(C_8:C_2):C_2$ |
Character table for $(C_8:C_2):C_2$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 8 sibling: | data not computed |
Degree 16 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 | ${\href{/padicField/2.8.0.1}{8} }$ | ${\href{/padicField/3.8.0.1}{8} }$ | R | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.8.7.2 | $x^{8} + 5$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
\(131\) | $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
131.2.1.2 | $x^{2} + 131$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
131.2.0.1 | $x^{2} + 127 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
131.2.1.1 | $x^{2} + 262$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.655.2t1.a.a | $1$ | $ 5 \cdot 131 $ | \(\Q(\sqrt{-655}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.131.2t1.a.a | $1$ | $ 131 $ | \(\Q(\sqrt{-131}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.655.4t1.a.a | $1$ | $ 5 \cdot 131 $ | 4.4.2145125.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
* | 1.5.4t1.a.a | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
1.655.4t1.a.b | $1$ | $ 5 \cdot 131 $ | 4.4.2145125.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
* | 1.5.4t1.a.b | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
2.655.4t3.b.a | $2$ | $ 5 \cdot 131 $ | 4.0.85805.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.3275.4t3.c.a | $2$ | $ 5^{2} \cdot 131 $ | 4.0.2145125.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 4.10725625.8t16.a.a | $4$ | $ 5^{4} \cdot 131^{2}$ | 8.0.1340703125.1 | $(C_8:C_2):C_2$ (as 8T16) | $1$ | $0$ |