Normalized defining polynomial
\( x^{8} + 2x^{6} - 2x^{5} - 16x^{4} - 2x^{3} - 16x^{2} + 17x - 3 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(205213530025\) \(\medspace = 5^{2}\cdot 7^{4}\cdot 43^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(25.94\) | 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: | $5^{2/3}7^{1/2}43^{1/2}\approx 50.729811745643495$ | ||
Ramified primes: | \(5\), \(7\), \(43\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{547}a^{7}-\frac{73}{547}a^{6}-\frac{139}{547}a^{5}-\frac{248}{547}a^{4}+\frac{37}{547}a^{3}+\frac{32}{547}a^{2}-\frac{164}{547}a-\frac{45}{547}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $\frac{492}{547}a^{7}+\frac{186}{547}a^{6}+\frac{1081}{547}a^{5}-\frac{582}{547}a^{4}-\frac{8052}{547}a^{3}-\frac{3948}{547}a^{2}-\frac{9578}{547}a+\frac{4663}{547}$, $\frac{186}{547}a^{7}+\frac{97}{547}a^{6}+\frac{402}{547}a^{5}-\frac{180}{547}a^{4}-\frac{2964}{547}a^{3}-\frac{1706}{547}a^{2}-\frac{3154}{547}a+\frac{2570}{547}$, $\frac{89}{547}a^{7}+\frac{67}{547}a^{6}+\frac{210}{547}a^{5}-\frac{192}{547}a^{4}-\frac{1630}{547}a^{3}-\frac{981}{547}a^{2}-\frac{1468}{547}a+\frac{1465}{547}$, $\frac{82}{547}a^{7}+\frac{31}{547}a^{6}+\frac{89}{547}a^{5}-\frac{97}{547}a^{4}-\frac{1342}{547}a^{3}-\frac{658}{547}a^{2}-\frac{867}{547}a+\frac{139}{547}$, $\frac{15}{547}a^{7}-\frac{1}{547}a^{6}+\frac{103}{547}a^{5}-\frac{438}{547}a^{4}+\frac{8}{547}a^{3}-\frac{614}{547}a^{2}+\frac{275}{547}a-\frac{128}{547}$ | 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: | \( 1406.67642757 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 1406.67642757 \cdot 1}{2\cdot\sqrt{205213530025}}\cr\approx \mathstrut & 0.980710754924 \end{aligned}\]
Galois group
$\PGOPlus(4,3)$ (as 8T45):
A solvable group of order 576 |
The 16 conjugacy class representatives for $(A_4\wr C_2):C_2$ |
Character table for $(A_4\wr C_2):C_2$ |
Intermediate fields
\(\Q(\sqrt{301}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Degree 36 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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | R | R | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | R | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(7\) | 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(43\) | 43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
43.6.3.2 | $x^{6} + 131 x^{4} + 80 x^{3} + 5548 x^{2} - 10240 x + 77452$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.43.2t1.a.a | $1$ | $ 43 $ | \(\Q(\sqrt{-43}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.301.2t1.a.a | $1$ | $ 7 \cdot 43 $ | \(\Q(\sqrt{301}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.175.3t2.b.a | $2$ | $ 5^{2} \cdot 7 $ | 3.1.175.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.52675.6t3.a.a | $2$ | $ 5^{2} \cdot 7^{2} \cdot 43 $ | 6.2.17044313125.2 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
2.1075.3t2.a.a | $2$ | $ 5^{2} \cdot 43 $ | 3.1.1075.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.323575.6t3.d.a | $2$ | $ 5^{2} \cdot 7 \cdot 43^{2}$ | 6.2.17044313125.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
4.2265025.6t9.a.a | $4$ | $ 5^{2} \cdot 7^{2} \cdot 43^{2}$ | 6.2.681772525.1 | $S_3^2$ (as 6T9) | $1$ | $0$ | |
* | 6.681772525.8t45.a.a | $6$ | $ 5^{2} \cdot 7^{3} \cdot 43^{3}$ | 8.4.205213530025.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $2$ |
6.681772525.12t161.a.a | $6$ | $ 5^{2} \cdot 7^{3} \cdot 43^{3}$ | 8.4.205213530025.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $-2$ | |
9.146...875.18t185.a.a | $9$ | $ 5^{6} \cdot 7^{6} \cdot 43^{3}$ | 8.4.205213530025.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $-1$ | |
9.426107828125.12t165.a.a | $9$ | $ 5^{6} \cdot 7^{3} \cdot 43^{3}$ | 8.4.205213530025.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $1$ | |
9.338...375.18t185.a.a | $9$ | $ 5^{6} \cdot 7^{3} \cdot 43^{6}$ | 8.4.205213530025.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $-1$ | |
9.116...625.18t179.a.a | $9$ | $ 5^{6} \cdot 7^{6} \cdot 43^{6}$ | 8.4.205213530025.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $1$ | |
12.726...625.24t1503.a.a | $12$ | $ 5^{10} \cdot 7^{6} \cdot 43^{6}$ | 8.4.205213530025.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $0$ |