Normalized defining polynomial
\( x^{8} - x^{7} - 8x^{6} + 6x^{5} + 17x^{4} - 13x^{3} + x^{2} + 17x - 10 \)
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: | \(186638688289\) \(\medspace = 41^{4}\cdot 257^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(25.64\) | 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: | $41^{1/2}257^{1/2}\approx 102.6498904042279$ | ||
Ramified primes: | \(41\), \(257\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$
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: | $a^{2}+a-1$, $a^{2}-a-3$, $\frac{1}{2}a^{7}-\frac{3}{2}a^{6}-\frac{3}{2}a^{5}+6a^{4}-\frac{3}{2}a^{3}-3a^{2}+6a-3$, $a^{7}-5a^{5}-a^{4}+2a^{3}-5a^{2}+1$, $a^{7}-\frac{5}{2}a^{6}-\frac{9}{2}a^{5}+\frac{27}{2}a^{4}-2a^{3}-\frac{29}{2}a^{2}+22a-9$ | 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: | \( 1965.77880514 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 1965.77880514 \cdot 1}{2\cdot\sqrt{186638688289}}\cr\approx \mathstrut & 1.43708856989 \end{aligned}\]
Galois group
$C_2^3:S_4$ (as 8T41):
A solvable group of order 192 |
The 14 conjugacy class representatives for $V_4^2:(S_3\times C_2)$ |
Character table for $V_4^2:(S_3\times C_2)$ |
Intermediate fields
\(\Q(\sqrt{41}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 sibling: | data not computed |
Degree 12 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 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 | ${\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(41\) | 41.8.4.1 | $x^{8} + 3772 x^{7} + 5335658 x^{6} + 3354711230 x^{5} + 791201413052 x^{4} + 137665893766 x^{3} + 86143602389 x^{2} + 18207949968812 x + 5534365087156$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(257\) | Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
Deg $4$ | $2$ | $2$ | $2$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.41.2t1.a.a | $1$ | $ 41 $ | \(\Q(\sqrt{41}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.10537.2t1.a.a | $1$ | $ 41 \cdot 257 $ | \(\Q(\sqrt{10537}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.257.2t1.a.a | $1$ | $ 257 $ | \(\Q(\sqrt{257}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.432017.6t3.b.a | $2$ | $ 41^{2} \cdot 257 $ | 6.6.4552163129.1 | $D_{6}$ (as 6T3) | $1$ | $2$ | |
2.257.3t2.a.a | $2$ | $ 257 $ | 3.3.257.1 | $S_3$ (as 3T2) | $1$ | $2$ | |
3.66049.6t8.a.a | $3$ | $ 257^{2}$ | 4.0.257.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.17712697.6t11.a.a | $3$ | $ 41^{3} \cdot 257 $ | 6.2.1169905924153.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
3.4552163129.6t11.a.a | $3$ | $ 41^{3} \cdot 257^{2}$ | 6.2.1169905924153.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
3.257.4t5.a.a | $3$ | $ 257 $ | 4.0.257.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
6.116...153.12t108.a.a | $6$ | $ 41^{3} \cdot 257^{3}$ | 8.4.186638688289.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $-2$ | |
* | 6.4552163129.8t41.a.a | $6$ | $ 41^{3} \cdot 257^{2}$ | 8.4.186638688289.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $2$ |
6.300...321.12t108.a.a | $6$ | $ 41^{3} \cdot 257^{4}$ | 8.4.186638688289.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $2$ | |
6.116...153.8t41.a.a | $6$ | $ 41^{3} \cdot 257^{3}$ | 8.4.186638688289.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $-2$ |