Normalized defining polynomial
\( x^{8} - 28x^{6} - 112x^{5} - 210x^{4} - 224x^{3} - 140x^{2} - 48x + 823552 \)
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: | \(1277992348243533546275162547509832257536\) \(\medspace = 2^{12}\cdot 11^{6}\cdot 74869^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(77\,324.31\) | 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: | $2^{11/4}11^{6/7}74869^{6/7}\approx 791458.158611407$ | ||
Ramified primes: | \(2\), \(11\), \(74869\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{7}-\frac{1}{4}a^{5}-\frac{1}{8}a^{3}+\frac{1}{4}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (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: | $\frac{30\!\cdots\!91}{4}a^{7}+15\!\cdots\!66a^{6}-33\!\cdots\!72a^{5}-21\!\cdots\!27a^{4}-\frac{56\!\cdots\!41}{2}a^{3}+50\!\cdots\!00a^{2}+19\!\cdots\!61a-44\!\cdots\!67$, $\frac{81\!\cdots\!79}{8}a^{7}-13\!\cdots\!28a^{6}+\frac{16\!\cdots\!13}{2}a^{5}+47\!\cdots\!33a^{4}-\frac{72\!\cdots\!11}{4}a^{3}-33\!\cdots\!11a^{2}+\frac{14\!\cdots\!57}{2}a-24\!\cdots\!55$, $77\!\cdots\!41a^{7}+\frac{32\!\cdots\!27}{2}a^{6}-\frac{67\!\cdots\!73}{2}a^{5}-21\!\cdots\!18a^{4}-37\!\cdots\!06a^{3}+50\!\cdots\!99a^{2}+20\!\cdots\!64a-44\!\cdots\!27$ (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: | \( 192707882917000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 192707882917000000 \cdot 1}{2\cdot\sqrt{1277992348243533546275162547509832257536}}\cr\approx \mathstrut & 4.20073060856289 \end{aligned}\] (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 | ${\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
2.4.9.1 | $x^{4} + 10 x^{2} + 2$ | $4$ | $1$ | $9$ | $D_{4}$ | $[2, 3, 7/2]$ | |
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
11.7.6.1 | $x^{7} + 11$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
\(74869\) | $\Q_{74869}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $7$ | $7$ | $1$ | $6$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 7.127...536.8t49.a.a | $7$ | $ 2^{12} \cdot 11^{6} \cdot 74869^{6}$ | 8.0.1277992348243533546275162547509832257536.1 | $A_8$ (as 8T49) | $1$ | $-1$ |
14.167...104.15t72.a.a | $14$ | $ 2^{34} \cdot 11^{12} \cdot 74869^{12}$ | 8.0.1277992348243533546275162547509832257536.1 | $A_8$ (as 8T49) | $1$ | $6$ | |
20.213...744.28t433.a.a | $20$ | $ 2^{46} \cdot 11^{18} \cdot 74869^{18}$ | 8.0.1277992348243533546275162547509832257536.1 | $A_8$ (as 8T49) | $1$ | $4$ | |
21.218...856.56.a.a | $21$ | $ 2^{56} \cdot 11^{18} \cdot 74869^{18}$ | 8.0.1277992348243533546275162547509832257536.1 | $A_8$ (as 8T49) | $1$ | $-3$ | |
21.218...856.336.a.a | $21$ | $ 2^{56} \cdot 11^{18} \cdot 74869^{18}$ | 8.0.1277992348243533546275162547509832257536.1 | $A_8$ (as 8T49) | $0$ | $-3$ | |
21.218...856.336.a.b | $21$ | $ 2^{56} \cdot 11^{18} \cdot 74869^{18}$ | 8.0.1277992348243533546275162547509832257536.1 | $A_8$ (as 8T49) | $0$ | $-3$ | |
28.279...816.56.a.a | $28$ | $ 2^{68} \cdot 11^{24} \cdot 74869^{24}$ | 8.0.1277992348243533546275162547509832257536.1 | $A_8$ (as 8T49) | $1$ | $-4$ | |
35.959...456.70.a.a | $35$ | $ 2^{108} \cdot 11^{30} \cdot 74869^{30}$ | 8.0.1277992348243533546275162547509832257536.1 | $A_8$ (as 8T49) | $1$ | $3$ | |
45.701...936.336.a.a | $45$ | $ 2^{130} \cdot 11^{39} \cdot 74869^{39}$ | 8.0.1277992348243533546275162547509832257536.1 | $A_8$ (as 8T49) | $0$ | $-3$ | |
45.701...936.336.a.b | $45$ | $ 2^{130} \cdot 11^{39} \cdot 74869^{39}$ | 8.0.1277992348243533546275162547509832257536.1 | $A_8$ (as 8T49) | $0$ | $-3$ | |
56.205...264.105.a.a | $56$ | $ 2^{154} \cdot 11^{48} \cdot 74869^{48}$ | 8.0.1277992348243533546275162547509832257536.1 | $A_8$ (as 8T49) | $1$ | $8$ | |
64.268...096.168.a.a | $64$ | $ 2^{176} \cdot 11^{54} \cdot 74869^{54}$ | 8.0.1277992348243533546275162547509832257536.1 | $A_8$ (as 8T49) | $1$ | $0$ | |
70.343...456.120.a.a | $70$ | $ 2^{188} \cdot 11^{60} \cdot 74869^{60}$ | 8.0.1277992348243533546275162547509832257536.1 | $A_8$ (as 8T49) | $1$ | $-2$ |