Normalized defining polynomial
\( x^{8} - 8x^{6} - 8x^{5} + 27x^{4} + 32x^{3} - 28x^{2} - 44x - 4 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-45091246208\) \(\medspace = -\,2^{7}\cdot 137^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(21.47\) | 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: | $2^{25/12}137^{1/2}\approx 49.60279034383193$ | ||
Ramified primes: | \(2\), \(137\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{88}a^{7}-\frac{3}{88}a^{6}+\frac{1}{88}a^{5}-\frac{1}{8}a^{4}-\frac{7}{22}a^{3}+\frac{7}{22}a^{2}-\frac{3}{11}a+\frac{7}{22}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | 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: | $\frac{1}{88}a^{7}-\frac{3}{88}a^{6}+\frac{1}{88}a^{5}+\frac{3}{8}a^{4}-\frac{7}{22}a^{3}-\frac{13}{11}a^{2}+\frac{8}{11}a+\frac{29}{22}$, $\frac{3}{88}a^{7}-\frac{9}{88}a^{6}-\frac{41}{88}a^{5}+\frac{1}{8}a^{4}+\frac{28}{11}a^{3}+\frac{16}{11}a^{2}-\frac{42}{11}a-\frac{89}{22}$, $\frac{3}{88}a^{7}-\frac{9}{88}a^{6}+\frac{3}{88}a^{5}-\frac{3}{8}a^{4}+\frac{1}{22}a^{3}+\frac{43}{22}a^{2}+\frac{13}{11}a-\frac{23}{22}$, $\frac{19}{44}a^{7}+\frac{31}{44}a^{6}-\frac{113}{44}a^{5}-\frac{31}{4}a^{4}-\frac{1}{11}a^{3}+\frac{166}{11}a^{2}+\frac{139}{11}a+\frac{12}{11}$ | 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: | \( 663.051239208 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 663.051239208 \cdot 1}{2\cdot\sqrt{45091246208}}\cr\approx \mathstrut & 1.54906832141 \end{aligned}\]
Galois group
$S_4\wr C_2$ (as 8T47):
A solvable group of order 1152 |
The 20 conjugacy class representatives for $S_4\wr C_2$ |
Character table for $S_4\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{137}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Degree 36 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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.3 | $x^{2} + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
2.4.4.5 | $x^{4} + 2 x + 2$ | $4$ | $1$ | $4$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
\(137\) | 137.8.4.1 | $x^{8} + 40552 x^{7} + 616674814 x^{6} + 4167912520178 x^{5} + 10563701578684025 x^{4} + 575230492397974 x^{3} + 10971113842265417 x^{2} + 1003833647367390048 x + 45253593243784800$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.1096.2t1.b.a | $1$ | $ 2^{3} \cdot 137 $ | \(\Q(\sqrt{-274}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.137.2t1.a.a | $1$ | $ 137 $ | \(\Q(\sqrt{137}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.1096.4t3.c.a | $2$ | $ 2^{3} \cdot 137 $ | 4.0.8768.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.35072.6t13.b.a | $4$ | $ 2^{8} \cdot 137 $ | 6.0.280576.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.9609728.12t34.b.a | $4$ | $ 2^{9} \cdot 137^{2}$ | 6.0.280576.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.658266368.12t34.b.a | $4$ | $ 2^{8} \cdot 137^{3}$ | 6.0.280576.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.600608.6t13.b.a | $4$ | $ 2^{5} \cdot 137^{2}$ | 6.0.280576.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.10532261888.12t201.a.a | $6$ | $ 2^{12} \cdot 137^{3}$ | 8.2.45091246208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
6.84258095104.12t202.a.a | $6$ | $ 2^{15} \cdot 137^{3}$ | 8.2.45091246208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
* | 6.329133184.8t47.a.a | $6$ | $ 2^{7} \cdot 137^{3}$ | 8.2.45091246208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
6.2633065472.12t200.a.a | $6$ | $ 2^{10} \cdot 137^{3}$ | 8.2.45091246208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.337032380416.16t1294.a.a | $9$ | $ 2^{17} \cdot 137^{3}$ | 8.2.45091246208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
9.107...312.18t272.a.a | $9$ | $ 2^{22} \cdot 137^{3}$ | 8.2.45091246208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.277...136.18t273.a.a | $9$ | $ 2^{22} \cdot 137^{6}$ | 8.2.45091246208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.866...848.18t274.a.a | $9$ | $ 2^{17} \cdot 137^{6}$ | 8.2.45091246208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
12.887...352.36t1763.a.a | $12$ | $ 2^{27} \cdot 137^{6}$ | 8.2.45091246208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.554...272.24t2821.a.a | $12$ | $ 2^{23} \cdot 137^{6}$ | 8.2.45091246208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.934...576.36t1758.a.a | $18$ | $ 2^{39} \cdot 137^{9}$ | 8.2.45091246208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |