Normalized defining polynomial
\( x^{8} - 28x^{6} - 112x^{5} - 210x^{4} - 224x^{3} - 140x^{2} - 48x + 823585 \)
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: | \(133100753213221593424899389161209856\) \(\medspace = 2^{28}\cdot 7^{8}\cdot 11^{6}\cdot 191^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(24\,576.64\) | 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: | $2^{61/16}7^{26/21}11^{6/7}191^{6/7}\approx 110098.77243012987$ | ||
Ramified primes: | \(2\), \(7\), \(11\), \(191\) | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{14}a^{5}-\frac{3}{14}a^{4}-\frac{1}{14}a^{3}-\frac{3}{14}a^{2}-\frac{3}{7}a$, $\frac{1}{28}a^{6}-\frac{3}{28}a^{4}-\frac{3}{14}a^{3}+\frac{13}{28}a^{2}+\frac{5}{14}a-\frac{1}{4}$, $\frac{1}{28}a^{7}-\frac{1}{28}a^{5}+\frac{1}{14}a^{4}-\frac{3}{28}a^{3}-\frac{5}{14}a^{2}-\frac{5}{28}a$
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: | $\frac{52\!\cdots\!23}{2}a^{7}-\frac{66\!\cdots\!53}{7}a^{6}-\frac{40\!\cdots\!65}{7}a^{5}+\frac{86\!\cdots\!82}{7}a^{4}-\frac{29\!\cdots\!83}{14}a^{3}+13\!\cdots\!29a^{2}-\frac{59\!\cdots\!71}{7}a+46\!\cdots\!06$, $\frac{42\!\cdots\!13}{28}a^{7}-\frac{28\!\cdots\!31}{28}a^{6}+\frac{45\!\cdots\!23}{28}a^{5}-\frac{52\!\cdots\!07}{28}a^{4}+\frac{12\!\cdots\!55}{28}a^{3}-\frac{88\!\cdots\!61}{28}a^{2}-\frac{37\!\cdots\!79}{28}a+\frac{62\!\cdots\!53}{4}$, $\frac{35\!\cdots\!05}{14}a^{7}-\frac{21\!\cdots\!19}{14}a^{6}+\frac{55\!\cdots\!38}{7}a^{5}-\frac{29\!\cdots\!15}{14}a^{4}+\frac{15\!\cdots\!73}{14}a^{3}+\frac{38\!\cdots\!87}{14}a^{2}-36\!\cdots\!72a+\frac{60\!\cdots\!89}{2}$ (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: | \( 102952955516000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 102952955516000 \cdot 4}{2\cdot\sqrt{133100753213221593424899389161209856}}\cr\approx \mathstrut & 0.879625946084417 \end{aligned}\] (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 | R | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.28.43 | $x^{8} + 8 x^{5} + 26$ | $8$ | $1$ | $28$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $[2, 3, 7/2, 4, 17/4]^{2}$ |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.7.8.3 | $x^{7} + 28 x^{2} + 7$ | $7$ | $1$ | $8$ | $C_7:C_3$ | $[4/3]_{3}$ | |
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
11.7.6.1 | $x^{7} + 11$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
\(191\) | $\Q_{191}$ | $x + 172$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
191.7.6.1 | $x^{7} + 191$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 7.133...856.8t49.a.a | $7$ | $ 2^{28} \cdot 7^{8} \cdot 11^{6} \cdot 191^{6}$ | 8.0.133100753213221593424899389161209856.1 | $A_8$ (as 8T49) | $1$ | $-1$ |
14.270...576.15t72.a.a | $14$ | $ 2^{40} \cdot 7^{16} \cdot 11^{12} \cdot 191^{12}$ | 8.0.133100753213221593424899389161209856.1 | $A_8$ (as 8T49) | $1$ | $6$ | |
20.176...744.28t433.a.a | $20$ | $ 2^{68} \cdot 7^{26} \cdot 11^{18} \cdot 191^{18}$ | 8.0.133100753213221593424899389161209856.1 | $A_8$ (as 8T49) | $1$ | $4$ | |
21.288...696.56.a.a | $21$ | $ 2^{82} \cdot 7^{26} \cdot 11^{18} \cdot 191^{18}$ | 8.0.133100753213221593424899389161209856.1 | $A_8$ (as 8T49) | $1$ | $-3$ | |
21.288...696.336.a.a | $21$ | $ 2^{82} \cdot 7^{26} \cdot 11^{18} \cdot 191^{18}$ | 8.0.133100753213221593424899389161209856.1 | $A_8$ (as 8T49) | $0$ | $-3$ | |
21.288...696.336.a.b | $21$ | $ 2^{82} \cdot 7^{26} \cdot 11^{18} \cdot 191^{18}$ | 8.0.133100753213221593424899389161209856.1 | $A_8$ (as 8T49) | $0$ | $-3$ | |
28.384...776.56.a.a | $28$ | $ 2^{110} \cdot 7^{34} \cdot 11^{24} \cdot 191^{24}$ | 8.0.133100753213221593424899389161209856.1 | $A_8$ (as 8T49) | $1$ | $-4$ | |
35.319...016.70.a.a | $35$ | $ 2^{134} \cdot 7^{42} \cdot 11^{30} \cdot 191^{30}$ | 8.0.133100753213221593424899389161209856.1 | $A_8$ (as 8T49) | $1$ | $3$ | |
45.761...136.336.a.a | $45$ | $ 2^{176} \cdot 7^{56} \cdot 11^{39} \cdot 191^{39}$ | 8.0.133100753213221593424899389161209856.1 | $A_8$ (as 8T49) | $0$ | $-3$ | |
45.761...136.336.a.b | $45$ | $ 2^{176} \cdot 7^{56} \cdot 11^{39} \cdot 191^{39}$ | 8.0.133100753213221593424899389161209856.1 | $A_8$ (as 8T49) | $0$ | $-3$ | |
56.276...296.105.a.a | $56$ | $ 2^{202} \cdot 7^{70} \cdot 11^{48} \cdot 191^{48}$ | 8.0.133100753213221593424899389161209856.1 | $A_8$ (as 8T49) | $1$ | $8$ | |
64.295...816.168.a.a | $64$ | $ 2^{244} \cdot 7^{80} \cdot 11^{54} \cdot 191^{54}$ | 8.0.133100753213221593424899389161209856.1 | $A_8$ (as 8T49) | $1$ | $0$ | |
70.801...704.120.a.a | $70$ | $ 2^{272} \cdot 7^{86} \cdot 11^{60} \cdot 191^{60}$ | 8.0.133100753213221593424899389161209856.1 | $A_8$ (as 8T49) | $1$ | $-2$ |