Normalized defining polynomial
\( x^{8} - x^{7} + 7x^{6} - 28x^{5} + 70x^{4} - 112x^{3} + 112x^{2} - 64x + 1 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-9380669484375\) \(\medspace = -\,3^{6}\cdot 5^{6}\cdot 7^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(41.83\) | 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: | $3^{6/7}5^{6/7}7^{7/8}\approx 55.91637477676722$ | ||
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: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3615}a^{7}+\frac{451}{3615}a^{6}+\frac{473}{1205}a^{5}+\frac{301}{723}a^{4}+\frac{142}{723}a^{3}-\frac{309}{1205}a^{2}+\frac{448}{3615}a-\frac{8}{3615}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $4$ | 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)
| |
Fundamental units: | $a$, $a^{2}-2a+2$, $\frac{117}{241}a^{7}-\frac{12}{241}a^{6}+\frac{697}{241}a^{5}-\frac{2737}{241}a^{4}+\frac{4745}{241}a^{3}-\frac{6516}{241}a^{2}+\frac{3734}{241}a+\frac{269}{241}$, $a^{2}-2$ | 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: | \( 3762.85592164 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 3762.85592164 \cdot 3}{2\cdot\sqrt{9380669484375}}\cr\approx \mathstrut & 1.82848553031 \end{aligned}\]
Galois group
$\PGL(2,7)$ (as 8T43):
A non-solvable group of order 336 |
The 9 conjugacy class representatives for $\PGL(2,7)$ |
Character table for $\PGL(2,7)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 14 sibling: | deg 14 |
Degree 16 sibling: | deg 16 |
Degree 21 sibling: | deg 21 |
Degree 24 sibling: | deg 24 |
Degree 28 siblings: | deg 28, deg 28 |
Degree 42 siblings: | deg 42, some 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 | ${\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | R | R | ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.7.6.1 | $x^{7} + 3$ | $7$ | $1$ | $6$ | $F_7$ | $[\ ]_{7}^{6}$ | |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.7.6.1 | $x^{7} + 5$ | $7$ | $1$ | $6$ | $F_7$ | $[\ ]_{7}^{6}$ | |
\(7\) | 7.8.7.2 | $x^{8} + 21$ | $8$ | $1$ | $7$ | $D_{8}$ | $[\ ]_{8}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
6.191442234375.42t82.a.a | $6$ | $ 3^{6} \cdot 5^{6} \cdot 7^{5}$ | 8.2.9380669484375.1 | $\PGL(2,7)$ (as 8T43) | $1$ | $0$ | |
6.191442234375.14t16.a.a | $6$ | $ 3^{6} \cdot 5^{6} \cdot 7^{5}$ | 8.2.9380669484375.1 | $\PGL(2,7)$ (as 8T43) | $1$ | $0$ | |
6.191442234375.14t16.a.b | $6$ | $ 3^{6} \cdot 5^{6} \cdot 7^{5}$ | 8.2.9380669484375.1 | $\PGL(2,7)$ (as 8T43) | $1$ | $0$ | |
* | 7.938...375.8t43.a.a | $7$ | $ 3^{6} \cdot 5^{6} \cdot 7^{7}$ | 8.2.9380669484375.1 | $\PGL(2,7)$ (as 8T43) | $1$ | $1$ |
7.134...625.16t713.a.a | $7$ | $ 3^{6} \cdot 5^{6} \cdot 7^{6}$ | 8.2.9380669484375.1 | $\PGL(2,7)$ (as 8T43) | $1$ | $-1$ | |
8.938...375.42t81.a.a | $8$ | $ 3^{6} \cdot 5^{6} \cdot 7^{7}$ | 8.2.9380669484375.1 | $\PGL(2,7)$ (as 8T43) | $1$ | $-2$ | |
8.938...375.21t20.a.a | $8$ | $ 3^{6} \cdot 5^{6} \cdot 7^{7}$ | 8.2.9380669484375.1 | $\PGL(2,7)$ (as 8T43) | $1$ | $2$ |