Normalized defining polynomial
\( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 49x^{4} - 56x^{3} + 63x^{2} - 36x + 32 \)
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)
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Discriminant: | \(65477433301\) \(\medspace = 7^{7}\cdot 43^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(22.49\) | 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: | $7^{47/42}43^{1/2}\approx 57.86809387704$ | ||
Ramified primes: | \(7\), \(43\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{301}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a$
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)
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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)
| |
Fundamental units: | $\frac{1}{2}a^{7}-\frac{3}{2}a^{6}+\frac{7}{2}a^{5}-\frac{9}{2}a^{4}+\frac{11}{2}a^{3}-\frac{5}{2}a^{2}+\frac{5}{2}a+1$, $\frac{1}{2}a^{7}-\frac{3}{2}a^{6}+4a^{5}-6a^{4}+8a^{3}-\frac{11}{2}a^{2}+6a-1$, $\frac{1}{2}a^{7}-\frac{7}{2}a^{6}+\frac{19}{2}a^{5}-\frac{39}{2}a^{4}+\frac{53}{2}a^{3}-\frac{63}{2}a^{2}+\frac{41}{2}a-19$ | 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: | \( 444.357525087 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 444.357525087 \cdot 1}{2\cdot\sqrt{65477433301}}\cr\approx \mathstrut & 1.35324425984 \end{aligned}\]
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: | 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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.7.7.1 | $x^{7} + 42 x + 7$ | $7$ | $1$ | $7$ | $F_7$ | $[7/6]_{6}$ | |
\(43\) | 43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
43.4.2.1 | $x^{4} + 84 x^{3} + 1856 x^{2} + 3864 x + 77452$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |