Normalized defining polynomial
\( x^{8} - 4x^{7} + 10x^{6} + 6x^{5} - 47x^{4} + 94x^{3} - 82x^{2} + 26x - 3 \)
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: | \(-1383025225728\) \(\medspace = -\,2^{12}\cdot 3\cdot 103^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(32.93\) | 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: | $2^{7/4}3^{1/2}103^{1/2}\approx 59.12644016162559$ | ||
Ramified primes: | \(2\), \(3\), \(103\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{243}a^{7}+\frac{40}{243}a^{6}+\frac{23}{81}a^{5}-\frac{13}{27}a^{4}-\frac{92}{243}a^{3}-\frac{22}{81}a^{2}-\frac{70}{243}a+\frac{35}{81}$
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: | $a-1$, $\frac{229}{243}a^{7}-\frac{884}{243}a^{6}+\frac{731}{81}a^{5}+\frac{182}{27}a^{4}-\frac{10619}{243}a^{3}+\frac{6761}{81}a^{2}-\frac{16192}{243}a+\frac{1211}{81}$, $\frac{254}{243}a^{7}-\frac{1261}{243}a^{6}+\frac{1225}{81}a^{5}-\frac{62}{27}a^{4}-\frac{14620}{243}a^{3}+\frac{13690}{81}a^{2}-\frac{35033}{243}a+\frac{2329}{81}$, $\frac{200}{81}a^{7}-\frac{802}{81}a^{6}+\frac{523}{27}a^{5}+\frac{253}{9}a^{4}-\frac{11191}{81}a^{3}+\frac{3736}{27}a^{2}-\frac{3659}{81}a+\frac{142}{27}$ | 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: | \( 2396.72439883 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 2396.72439883 \cdot 1}{2\cdot\sqrt{1383025225728}}\cr\approx \mathstrut & 1.01105003146 \end{aligned}\]
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
\(\Q(\sqrt{103}) \) |
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 | R | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.12.20 | $x^{8} - 2 x^{7} + 2 x^{6} + 8 x^{5} + 8 x^{4} - 4 x^{3} - 4 x^{2} + 4$ | $4$ | $2$ | $12$ | $C_2^3: C_4$ | $[2, 2, 2]^{4}$ |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(103\) | 103.8.4.1 | $x^{8} + 416 x^{6} + 176 x^{5} + 64080 x^{4} - 35904 x^{3} + 4330880 x^{2} - 5564416 x + 109124096$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.1236.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 103 $ | \(\Q(\sqrt{-309}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.412.2t1.a.a | $1$ | $ 2^{2} \cdot 103 $ | \(\Q(\sqrt{103}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.4944.4t3.c.a | $2$ | $ 2^{4} \cdot 3 \cdot 103 $ | 4.0.14832.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.14832.6t13.b.a | $4$ | $ 2^{4} \cdot 3^{2} \cdot 103 $ | 6.0.44496.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.18332352.12t34.b.a | $4$ | $ 2^{6} \cdot 3^{3} \cdot 103^{2}$ | 6.0.44496.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.2517643008.12t34.b.a | $4$ | $ 2^{8} \cdot 3^{2} \cdot 103^{3}$ | 6.0.44496.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.2036928.6t13.b.a | $4$ | $ 2^{6} \cdot 3 \cdot 103^{2}$ | 6.0.44496.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.362540593152.12t201.a.a | $6$ | $ 2^{12} \cdot 3^{4} \cdot 103^{3}$ | 8.2.1383025225728.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
6.271905444864.12t202.a.a | $6$ | $ 2^{10} \cdot 3^{5} \cdot 103^{3}$ | 8.2.1383025225728.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
* | 6.3356857344.8t47.a.a | $6$ | $ 2^{10} \cdot 3 \cdot 103^{3}$ | 8.2.1383025225728.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
6.40282288128.12t200.a.a | $6$ | $ 2^{12} \cdot 3^{2} \cdot 103^{3}$ | 8.2.1383025225728.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.120846864384.16t1294.a.a | $9$ | $ 2^{12} \cdot 3^{3} \cdot 103^{3}$ | 8.2.1383025225728.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
9.326...368.18t272.a.a | $9$ | $ 2^{12} \cdot 3^{6} \cdot 103^{3}$ | 8.2.1383025225728.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.228...304.18t273.c.a | $9$ | $ 2^{18} \cdot 3^{6} \cdot 103^{6}$ | 8.2.1383025225728.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.845...752.18t274.c.a | $9$ | $ 2^{18} \cdot 3^{3} \cdot 103^{6}$ | 8.2.1383025225728.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
12.109...592.36t1763.a.a | $12$ | $ 2^{22} \cdot 3^{7} \cdot 103^{6}$ | 8.2.1383025225728.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.121...288.24t2821.a.a | $12$ | $ 2^{22} \cdot 3^{5} \cdot 103^{6}$ | 8.2.1383025225728.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.110...944.36t1758.a.a | $18$ | $ 2^{32} \cdot 3^{9} \cdot 103^{9}$ | 8.2.1383025225728.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |