Normalized defining polynomial
\( x^{8} - 4x^{7} + 3217 \)
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: | \(72641749645773438449680384\) \(\medspace = 2^{16}\cdot 3217^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(1708.63\) | 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: | $2^{103/48}3217^{6/7}\approx 4491.038960544058$ | ||
Ramified primes: | \(2\), \(3217\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$ (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: | $\frac{30\!\cdots\!64}{3}a^{7}+\frac{13\!\cdots\!92}{3}a^{6}-\frac{51\!\cdots\!21}{3}a^{5}-\frac{24\!\cdots\!06}{3}a^{4}-\frac{34\!\cdots\!15}{3}a^{3}+\frac{75\!\cdots\!62}{3}a^{2}+\frac{46\!\cdots\!55}{3}a+\frac{78\!\cdots\!96}{3}$, $91\!\cdots\!04a^{7}-10\!\cdots\!86a^{6}+14\!\cdots\!08a^{5}+79\!\cdots\!52a^{4}+21\!\cdots\!10a^{3}+65\!\cdots\!62a^{2}-13\!\cdots\!08a-43\!\cdots\!71$, $\frac{14\!\cdots\!89}{3}a^{7}-\frac{70\!\cdots\!05}{3}a^{6}-\frac{24\!\cdots\!25}{3}a^{5}-\frac{86\!\cdots\!18}{3}a^{4}-\frac{30\!\cdots\!14}{3}a^{3}-\frac{10\!\cdots\!30}{3}a^{2}-\frac{37\!\cdots\!76}{3}a-\frac{13\!\cdots\!78}{3}$ (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: | \( 3669575090.38 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 3669575090.38 \cdot 2}{2\cdot\sqrt{72641749645773438449680384}}\cr\approx \mathstrut & 0.671030259141 \end{aligned}\] (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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.16.63 | $x^{8} + 2 x^{4} + 4 x + 6$ | $8$ | $1$ | $16$ | $V_4^2:(S_3\times C_2)$ | $[4/3, 4/3, 2, 7/3, 7/3]_{3}^{2}$ |
\(3217\) | $\Q_{3217}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $7$ | $7$ | $1$ | $6$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 7.726...384.8t49.a.a | $7$ | $ 2^{16} \cdot 3217^{6}$ | 8.0.72641749645773438449680384.1 | $A_8$ (as 8T49) | $1$ | $-1$ |
14.515...144.15t72.a.a | $14$ | $ 2^{22} \cdot 3217^{12}$ | 8.0.72641749645773438449680384.1 | $A_8$ (as 8T49) | $1$ | $6$ | |
20.374...296.28t433.a.a | $20$ | $ 2^{38} \cdot 3217^{18}$ | 8.0.72641749645773438449680384.1 | $A_8$ (as 8T49) | $1$ | $4$ | |
21.383...104.56.a.a | $21$ | $ 2^{48} \cdot 3217^{18}$ | 8.0.72641749645773438449680384.1 | $A_8$ (as 8T49) | $1$ | $-3$ | |
21.383...104.336.a.a | $21$ | $ 2^{48} \cdot 3217^{18}$ | 8.0.72641749645773438449680384.1 | $A_8$ (as 8T49) | $0$ | $-3$ | |
21.383...104.336.a.b | $21$ | $ 2^{48} \cdot 3217^{18}$ | 8.0.72641749645773438449680384.1 | $A_8$ (as 8T49) | $0$ | $-3$ | |
28.278...936.56.a.a | $28$ | $ 2^{64} \cdot 3217^{24}$ | 8.0.72641749645773438449680384.1 | $A_8$ (as 8T49) | $1$ | $-4$ | |
35.790...904.70.a.a | $35$ | $ 2^{72} \cdot 3217^{30}$ | 8.0.72641749645773438449680384.1 | $A_8$ (as 8T49) | $1$ | $3$ | |
45.782...528.336.a.a | $45$ | $ 2^{100} \cdot 3217^{39}$ | 8.0.72641749645773438449680384.1 | $A_8$ (as 8T49) | $0$ | $-3$ | |
45.782...528.336.a.b | $45$ | $ 2^{100} \cdot 3217^{39}$ | 8.0.72641749645773438449680384.1 | $A_8$ (as 8T49) | $0$ | $-3$ | |
56.118...336.105.a.a | $56$ | $ 2^{112} \cdot 3217^{48}$ | 8.0.72641749645773438449680384.1 | $A_8$ (as 8T49) | $1$ | $8$ | |
64.880...576.168.a.a | $64$ | $ 2^{138} \cdot 3217^{54}$ | 8.0.72641749645773438449680384.1 | $A_8$ (as 8T49) | $1$ | $0$ | |
70.159...296.120.a.a | $70$ | $ 2^{152} \cdot 3217^{60}$ | 8.0.72641749645773438449680384.1 | $A_8$ (as 8T49) | $1$ | $-2$ |