Normalized defining polynomial
\( x^{8} + 2x^{6} - 2x^{5} - 6x^{4} - 2x^{3} + 9x^{2} + 10x + 4 \)
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: | \(31510377\) \(\medspace = 3^{4}\cdot 73^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(8.66\) | 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: | $3^{1/2}73^{3/4}\approx 43.25663903061054$ | ||
Ramified primes: | \(3\), \(73\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{73}) \) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{16}a^{7}-\frac{3}{16}a^{6}+\frac{3}{16}a^{5}-\frac{3}{16}a^{4}-\frac{5}{16}a^{3}+\frac{5}{16}a^{2}+\frac{1}{8}a-\frac{1}{4}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | \( -\frac{9}{16} a^{7} + \frac{11}{16} a^{6} - \frac{27}{16} a^{5} + \frac{43}{16} a^{4} + \frac{13}{16} a^{3} - \frac{29}{16} a^{2} - \frac{25}{8} a + \frac{1}{4} \) (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: | $\frac{7}{16}a^{7}-\frac{5}{16}a^{6}+\frac{13}{16}a^{5}-\frac{21}{16}a^{4}-\frac{35}{16}a^{3}+\frac{27}{16}a^{2}+\frac{31}{8}a+\frac{5}{4}$, $\frac{15}{16}a^{7}-\frac{13}{16}a^{6}+\frac{37}{16}a^{5}-\frac{53}{16}a^{4}-\frac{59}{16}a^{3}+\frac{51}{16}a^{2}+\frac{43}{8}a+\frac{9}{4}$, $\frac{7}{16}a^{7}-\frac{13}{16}a^{6}+\frac{29}{16}a^{5}-\frac{53}{16}a^{4}+\frac{21}{16}a^{3}+\frac{11}{16}a^{2}+\frac{15}{8}a-\frac{3}{4}$ | 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: | \( 11.1813393987 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 11.1813393987 \cdot 1}{6\cdot\sqrt{31510377}}\cr\approx \mathstrut & 0.517410518971 \end{aligned}\]
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
\(\Q(\sqrt{-3}) \), 4.0.657.1 |
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: | 16.4.16695593025891398600698809.1, 16.0.28196723068685041735089.1 |
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 | ${\href{/padicField/2.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(73\) | 73.4.0.1 | $x^{4} + 16 x^{2} + 56 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
73.4.3.4 | $x^{4} + 803$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.73.2t1.a.a | $1$ | $ 73 $ | \(\Q(\sqrt{73}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.219.2t1.a.a | $1$ | $ 3 \cdot 73 $ | \(\Q(\sqrt{-219}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.219.4t1.a.a | $1$ | $ 3 \cdot 73 $ | 4.0.3501153.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.219.4t1.a.b | $1$ | $ 3 \cdot 73 $ | 4.0.3501153.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.73.4t1.a.a | $1$ | $ 73 $ | 4.4.389017.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.73.4t1.a.b | $1$ | $ 73 $ | 4.4.389017.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
2.15987.4t3.a.a | $2$ | $ 3 \cdot 73^{2}$ | 4.0.3501153.2 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 2.219.4t3.c.a | $2$ | $ 3 \cdot 73 $ | 4.0.657.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
* | 2.219.8t17.a.a | $2$ | $ 3 \cdot 73 $ | 8.0.31510377.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |
2.15987.8t17.a.a | $2$ | $ 3 \cdot 73^{2}$ | 8.0.31510377.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ | |
* | 2.219.8t17.a.b | $2$ | $ 3 \cdot 73 $ | 8.0.31510377.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |
2.15987.8t17.a.b | $2$ | $ 3 \cdot 73^{2}$ | 8.0.31510377.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |