Normalized defining polynomial
\( x^{8} - 4x^{7} + 5x^{6} - x^{5} - 7x^{4} + 11x^{3} - 6x^{2} + x + 1 \)
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: | \(88410125\) \(\medspace = 5^{3}\cdot 29^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(9.85\) | 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: | $5^{3/4}29^{1/2}\approx 18.006383777357115$ | ||
Ramified primes: | \(5\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7}a^{7}-\frac{2}{7}a^{5}-\frac{2}{7}a^{4}-\frac{1}{7}a^{3}+\frac{1}{7}a-\frac{2}{7}$
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: | $\frac{8}{7}a^{7}-4a^{6}+\frac{26}{7}a^{5}+\frac{5}{7}a^{4}-\frac{50}{7}a^{3}+8a^{2}-\frac{20}{7}a+\frac{5}{7}$, $a-1$, $a$, $\frac{1}{7}a^{7}-a^{6}+\frac{12}{7}a^{5}-\frac{2}{7}a^{4}-\frac{8}{7}a^{3}+3a^{2}-\frac{13}{7}a+\frac{5}{7}$, $\frac{2}{7}a^{7}-a^{6}+\frac{3}{7}a^{5}+\frac{10}{7}a^{4}-\frac{16}{7}a^{3}+2a^{2}+\frac{9}{7}a-\frac{4}{7}$ | 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: | \( 7.58920427421 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 7.58920427421 \cdot 1}{2\cdot\sqrt{88410125}}\cr\approx \mathstrut & 0.254914749114 \end{aligned}\]
Galois group
$C_4\wr C_2$ (as 8T17):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_4\wr C_2$ |
Character table for $C_4\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{29}) \), 4.4.4205.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 8 sibling: | data not computed |
Degree 16 siblings: | 16.0.145220537353515625.1, 16.8.4885218876572265625.1 |
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 | ${\href{/padicField/2.8.0.1}{8} }$ | ${\href{/padicField/3.8.0.1}{8} }$ | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
5.4.3.4 | $x^{4} + 15$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(29\) | 29.8.4.1 | $x^{8} + 2784 x^{7} + 2906616 x^{6} + 1348864734 x^{5} + 234834277018 x^{4} + 41857830864 x^{3} + 492109772617 x^{2} + 3561769809750 x + 616658760166$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.145.2t1.a.a | $1$ | $ 5 \cdot 29 $ | \(\Q(\sqrt{145}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.29.2t1.a.a | $1$ | $ 29 $ | \(\Q(\sqrt{29}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.5.4t1.a.a | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.145.4t1.b.a | $1$ | $ 5 \cdot 29 $ | 4.0.105125.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.5.4t1.a.b | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.145.4t1.b.b | $1$ | $ 5 \cdot 29 $ | 4.0.105125.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
2.725.4t3.c.a | $2$ | $ 5^{2} \cdot 29 $ | 4.0.105125.1 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
* | 2.145.4t3.a.a | $2$ | $ 5 \cdot 29 $ | 4.4.4205.1 | $D_{4}$ (as 4T3) | $1$ | $2$ |
2.725.8t17.b.a | $2$ | $ 5^{2} \cdot 29 $ | 8.4.88410125.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ | |
2.725.8t17.b.b | $2$ | $ 5^{2} \cdot 29 $ | 8.4.88410125.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ | |
* | 2.145.8t17.d.a | $2$ | $ 5 \cdot 29 $ | 8.4.88410125.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |
* | 2.145.8t17.d.b | $2$ | $ 5 \cdot 29 $ | 8.4.88410125.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |