Normalized defining polynomial
\( x^{8} - 4x^{7} + 8x^{6} - 10x^{5} + 11x^{4} - 6x^{3} - 4x^{2} + 4x + 2 \)
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: | \(282300416\) \(\medspace = 2^{12}\cdot 41^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.39\) | 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: | $2^{3/2}41^{3/4}\approx 45.82820910616052$ | ||
Ramified primes: | \(2\), \(41\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{41}) \) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{31}a^{7}-\frac{1}{31}a^{6}+\frac{5}{31}a^{5}+\frac{5}{31}a^{4}-\frac{5}{31}a^{3}+\frac{10}{31}a^{2}-\frac{5}{31}a-\frac{11}{31}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{12}{31} a^{7} + \frac{43}{31} a^{6} - \frac{91}{31} a^{5} + \frac{126}{31} a^{4} - \frac{157}{31} a^{3} + \frac{97}{31} a^{2} - \frac{2}{31} a - \frac{23}{31} \) (order $4$) | 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: | $\frac{1}{31}a^{7}-\frac{1}{31}a^{6}+\frac{5}{31}a^{5}+\frac{5}{31}a^{4}-\frac{5}{31}a^{3}+\frac{10}{31}a^{2}-\frac{5}{31}a-\frac{11}{31}$, $\frac{10}{31}a^{7}-\frac{41}{31}a^{6}+\frac{81}{31}a^{5}-\frac{105}{31}a^{4}+\frac{136}{31}a^{3}-\frac{117}{31}a^{2}+\frac{12}{31}a+\frac{45}{31}$, $\frac{3}{31}a^{7}-\frac{3}{31}a^{6}-\frac{16}{31}a^{5}+\frac{46}{31}a^{4}-\frac{77}{31}a^{3}+\frac{123}{31}a^{2}-\frac{108}{31}a+\frac{29}{31}$ | 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: | \( 39.2631690627 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 39.2631690627 \cdot 1}{4\cdot\sqrt{282300416}}\cr\approx \mathstrut & 0.910519126521 \end{aligned}\]
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
\(\Q(\sqrt{-1}) \), 4.0.656.1 |
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: | 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 | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.8 | $x^{4} + 2 x^{3} + 2$ | $4$ | $1$ | $6$ | $D_{4}$ | $[2, 2]^{2}$ |
2.4.6.8 | $x^{4} + 2 x^{3} + 2$ | $4$ | $1$ | $6$ | $D_{4}$ | $[2, 2]^{2}$ | |
\(41\) | $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
41.4.3.1 | $x^{4} + 41$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.41.2t1.a.a | $1$ | $ 41 $ | \(\Q(\sqrt{41}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.164.2t1.a.a | $1$ | $ 2^{2} \cdot 41 $ | \(\Q(\sqrt{-41}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.41.4t1.a.a | $1$ | $ 41 $ | 4.4.68921.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.41.4t1.a.b | $1$ | $ 41 $ | 4.4.68921.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.164.4t1.a.a | $1$ | $ 2^{2} \cdot 41 $ | 4.0.1102736.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.164.4t1.a.b | $1$ | $ 2^{2} \cdot 41 $ | 4.0.1102736.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
2.6724.4t3.b.a | $2$ | $ 2^{2} \cdot 41^{2}$ | 4.0.1102736.2 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 2.164.4t3.c.a | $2$ | $ 2^{2} \cdot 41 $ | 4.0.656.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
* | 2.656.8t17.b.a | $2$ | $ 2^{4} \cdot 41 $ | 8.0.282300416.2 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |
2.26896.8t17.b.a | $2$ | $ 2^{4} \cdot 41^{2}$ | 8.0.282300416.2 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ | |
* | 2.656.8t17.b.b | $2$ | $ 2^{4} \cdot 41 $ | 8.0.282300416.2 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |
2.26896.8t17.b.b | $2$ | $ 2^{4} \cdot 41^{2}$ | 8.0.282300416.2 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |