Normalized defining polynomial
\( x^{8} - 2x^{7} - 11x^{6} + 36x^{5} - x^{4} - 131x^{3} + 222x^{2} - 156x - 404 \)
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)
| |
Discriminant: | \(40608119002500\) \(\medspace = 2^{2}\cdot 3^{4}\cdot 5^{4}\cdot 7^{4}\cdot 17^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(50.24\) | 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: | $2^{2/3}3^{1/2}5^{1/2}7^{1/2}17^{1/2}\approx 67.06652032217488$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(7\), \(17\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{1312198}a^{7}+\frac{230877}{656099}a^{6}-\frac{60809}{1312198}a^{5}-\frac{253882}{656099}a^{4}+\frac{465055}{1312198}a^{3}-\frac{578449}{1312198}a^{2}-\frac{228364}{656099}a-\frac{215982}{656099}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{4}$, which has order $8$
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: | $\frac{11117}{1312198}a^{7}+\frac{321}{656099}a^{6}-\frac{231683}{1312198}a^{5}+\frac{131704}{656099}a^{4}+\frac{1268513}{1312198}a^{3}-\frac{2159531}{1312198}a^{2}-\frac{931656}{656099}a+\frac{906545}{656099}$, $\frac{16694}{656099}a^{7}+\frac{14125}{656099}a^{6}-\frac{160293}{656099}a^{5}+\frac{186864}{656099}a^{4}+\frac{664802}{656099}a^{3}-\frac{818623}{656099}a^{2}+\frac{1221445}{656099}a+\frac{2601489}{656099}$, $\frac{3906}{656099}a^{7}-\frac{5027}{656099}a^{6}-\frac{12116}{656099}a^{5}+\frac{61093}{656099}a^{4}-\frac{233301}{656099}a^{3}+\frac{183162}{656099}a^{2}-\frac{46387}{656099}a-\frac{420855}{656099}$, $\frac{2137}{1312198}a^{7}-\frac{2299}{656099}a^{6}-\frac{41231}{1312198}a^{5}+\frac{48039}{656099}a^{4}+\frac{488649}{1312198}a^{3}-\frac{54997}{1312198}a^{2}-\frac{1188410}{656099}a-\frac{1628135}{656099}$, $\frac{13112}{656099}a^{7}+\frac{36876}{656099}a^{6}-\frac{167323}{656099}a^{5}-\frac{365015}{656099}a^{4}+\frac{1329252}{656099}a^{3}+\frac{1849449}{656099}a^{2}-\frac{4338557}{656099}a-\frac{2433697}{656099}$ | 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: | \( 4664.45792275 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 4664.45792275 \cdot 8}{2\cdot\sqrt{40608119002500}}\cr\approx \mathstrut & 1.84941533279 \end{aligned}\]
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
\(\Q(\sqrt{1785}) \) |
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 | R | R | R | R | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | R | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.6.3.1 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(5\) | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(7\) | 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(17\) | 17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
17.6.3.2 | $x^{6} + 289 x^{2} - 68782$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.35.2t1.a.a | $1$ | $ 5 \cdot 7 $ | \(\Q(\sqrt{-35}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.51.2t1.a.a | $1$ | $ 3 \cdot 17 $ | \(\Q(\sqrt{-51}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.1785.2t1.a.a | $1$ | $ 3 \cdot 5 \cdot 7 \cdot 17 $ | \(\Q(\sqrt{1785}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.204.3t2.a.a | $2$ | $ 2^{2} \cdot 3 \cdot 17 $ | 3.1.204.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.364140.6t3.b.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17^{2}$ | 6.2.90998586000.3 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
2.140.3t2.a.a | $2$ | $ 2^{2} \cdot 5 \cdot 7 $ | 3.1.140.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.249900.6t3.b.a | $2$ | $ 2^{2} \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 17 $ | 6.2.90998586000.2 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
4.12744900.6t9.b.a | $4$ | $ 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 17^{2}$ | 6.2.22749646500.1 | $S_3^2$ (as 6T9) | $1$ | $0$ | |
* | 6.22749646500.8t45.a.a | $6$ | $ 2^{2} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 17^{3}$ | 8.4.40608119002500.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $2$ |
6.22749646500.12t161.a.a | $6$ | $ 2^{2} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 17^{3}$ | 8.4.40608119002500.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $-2$ | |
9.482...000.18t185.a.a | $9$ | $ 2^{6} \cdot 3^{6} \cdot 5^{3} \cdot 7^{3} \cdot 17^{6}$ | 8.4.40608119002500.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $-1$ | |
9.363994344000.12t165.a.a | $9$ | $ 2^{6} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 17^{3}$ | 8.4.40608119002500.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $1$ | |
9.156...000.18t185.a.a | $9$ | $ 2^{6} \cdot 3^{3} \cdot 5^{6} \cdot 7^{6} \cdot 17^{3}$ | 8.4.40608119002500.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $-1$ | |
9.207...000.18t179.a.a | $9$ | $ 2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 17^{6}$ | 8.4.40608119002500.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $1$ | |
12.331...000.24t1503.a.a | $12$ | $ 2^{10} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 17^{6}$ | 8.4.40608119002500.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $0$ |