Normalized defining polynomial
\( x^{8} - x^{6} - x^{4} + x^{2} + 1 \)
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: | \(3504384\) \(\medspace = 2^{8}\cdot 3^{4}\cdot 13^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(6.58\) | 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: | $2^{7/4}3^{1/2}13^{1/2}\approx 21.005585720479342$ | ||
Ramified primes: | \(2\), \(3\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$
Monogenic: | Yes | |
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: | \( -a^{6} + a^{4} \) (order $6$) | 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: | $a^{7}-a^{5}$, $a^{7}-2a^{5}+a^{4}-a^{2}+a$, $a^{7}-a^{5}-a^{4}+a^{2}+a$ | 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: | \( 1.89186954131 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 1.89186954131 \cdot 1}{6\cdot\sqrt{3504384}}\cr\approx \mathstrut & 0.262514661501 \end{aligned}\]
Galois group
$C_2\wr C_2^2$ (as 8T29):
A solvable group of order 64 |
The 16 conjugacy class representatives for $(((C_4 \times C_2): C_2):C_2):C_2$ |
Character table for $(((C_4 \times C_2): C_2):C_2):C_2$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 4.0.117.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | data not computed |
Degree 16 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 | R | R | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.3 | $x^{8} + 2 x^{7} + 32 x^{6} + 116 x^{5} + 456 x^{4} + 696 x^{3} + 1152 x^{2} + 432 x + 1296$ | $2$ | $4$ | $8$ | $C_2^3: C_4$ | $[2, 2, 2]^{4}$ |
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.39.2t1.a.a | $1$ | $ 3 \cdot 13 $ | \(\Q(\sqrt{-39}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.156.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 13 $ | \(\Q(\sqrt{39}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.52.2t1.a.a | $1$ | $ 2^{2} \cdot 13 $ | \(\Q(\sqrt{-13}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.624.4t3.d.a | $2$ | $ 2^{4} \cdot 3 \cdot 13 $ | 4.4.7488.1 | $D_{4}$ (as 4T3) | $1$ | $2$ | |
* | 2.39.4t3.a.a | $2$ | $ 3 \cdot 13 $ | 4.0.117.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
2.1872.4t3.h.a | $2$ | $ 2^{4} \cdot 3^{2} \cdot 13 $ | 4.0.7488.4 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.624.4t3.e.a | $2$ | $ 2^{4} \cdot 3 \cdot 13 $ | 4.0.7488.1 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
2.624.4t3.b.a | $2$ | $ 2^{4} \cdot 3 \cdot 13 $ | 4.0.1872.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.208.4t3.c.a | $2$ | $ 2^{4} \cdot 13 $ | 4.0.832.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.5061888.8t29.a.a | $4$ | $ 2^{8} \cdot 3^{2} \cdot 13^{3}$ | 8.0.3504384.1 | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $0$ | |
* | 4.29952.8t29.a.a | $4$ | $ 2^{8} \cdot 3^{2} \cdot 13 $ | 8.0.3504384.1 | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $0$ |