Normalized defining polynomial
\( x^{8} - 896x + 1568 \)
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: | \(3715492241427398656\) \(\medspace = 2^{28}\cdot 7^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(209.53\) | 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: | $2^{61/16}7^{12/7}\approx 394.83564736477246$ | ||
Ramified primes: | \(2\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{8}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{112}a^{7}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
Unit group
Rank: | $3$ | 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: | $\frac{1}{2}a^{7}-\frac{7}{2}a^{6}+14a^{5}-35a^{4}+56a^{3}-56a^{2}+4a+41$, $\frac{3657821}{56}a^{7}-\frac{11906707}{4}a^{6}+\frac{59100331}{4}a^{5}-\frac{208931515}{4}a^{4}+150822078a^{3}-359352165a^{2}+646374634a-520460383$, $\frac{88\!\cdots\!99}{112}a^{7}-\frac{11\!\cdots\!19}{8}a^{6}+\frac{50\!\cdots\!65}{4}a^{5}+\frac{21\!\cdots\!35}{4}a^{4}-\frac{74\!\cdots\!99}{2}a^{3}+14\!\cdots\!69a^{2}-46\!\cdots\!20a+51\!\cdots\!11$ (assuming GRH) | 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: | \( 468980.925517 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 468980.925517 \cdot 4}{2\cdot\sqrt{3715492241427398656}}\cr\approx \mathstrut & 0.758396760220 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 20160 |
The 14 conjugacy class representatives for $A_8$ |
Character table for $A_8$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 15 siblings: | deg 15, deg 15 |
Degree 28 sibling: | deg 28 |
Degree 35 sibling: | deg 35 |
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 | R | ${\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.28.62 | $x^{8} + 8 x^{5} + 2$ | $8$ | $1$ | $28$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $[2, 3, 7/2, 4, 17/4]^{2}$ |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.7.12.1 | $x^{7} + 42 x^{6} + 7$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ |