Normalized defining polynomial
\( x^{8} - 2x^{7} + 6x^{6} - 2x^{5} + 26x^{4} - 24x^{3} - 24x^{2} + 16x + 4 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(11027360000\) \(\medspace = 2^{8}\cdot 5^{4}\cdot 41^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.00\) | 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: | $2^{31/24}5^{1/2}41^{1/2}\approx 35.05155920981089$ | ||
Ramified primes: | \(2\), \(5\), \(41\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{41}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{110}a^{7}+\frac{1}{5}a^{6}-\frac{8}{55}a^{5}-\frac{1}{110}a^{4}+\frac{1}{55}a^{3}+\frac{12}{55}a^{2}+\frac{1}{55}a-\frac{23}{55}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $\frac{13}{5}a^{7}-\frac{14}{5}a^{6}+\frac{129}{10}a^{5}+\frac{69}{10}a^{4}+\frac{366}{5}a^{3}+\frac{27}{5}a^{2}-\frac{304}{5}a-\frac{63}{5}$, $\frac{27}{22}a^{7}-\frac{3}{2}a^{6}+\frac{70}{11}a^{5}+\frac{25}{11}a^{4}+\frac{379}{11}a^{3}-\frac{28}{11}a^{2}-\frac{303}{11}a-\frac{27}{11}$, $\frac{89}{22}a^{7}-\frac{9}{2}a^{6}+\frac{223}{11}a^{5}+\frac{219}{22}a^{4}+\frac{1255}{11}a^{3}+\frac{45}{11}a^{2}-\frac{1033}{11}a-\frac{210}{11}$, $a-1$, $\frac{207}{110}a^{7}-\frac{21}{10}a^{6}+\frac{1033}{110}a^{5}+\frac{254}{55}a^{4}+\frac{2902}{55}a^{3}+\frac{64}{55}a^{2}-\frac{2433}{55}a-\frac{471}{55}$ | 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: | \( 143.203954585 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 143.203954585 \cdot 1}{2\cdot\sqrt{11027360000}}\cr\approx \mathstrut & 0.430694135723 \end{aligned}\]
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
\(\Q(\sqrt{5}) \) |
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 | ${\href{/padicField/3.8.0.1}{8} }$ | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | R | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.12 | $x^{8} - 2 x^{6} - 2 x^{5} + 8 x^{4} + 8 x^{3} - 4 x + 4$ | $4$ | $2$ | $8$ | $A_4\wr C_2$ | $[4/3, 4/3, 4/3, 4/3]_{3}^{6}$ |
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(41\) | $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
41.2.1.2 | $x^{2} + 123$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
41.4.2.2 | $x^{4} - 45592 x^{3} - 53825497 x^{2} - 1482642 x + 10086$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.41.2t1.a.a | $1$ | $ 41 $ | \(\Q(\sqrt{41}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.205.2t1.a.a | $1$ | $ 5 \cdot 41 $ | \(\Q(\sqrt{205}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.205.4t3.c.a | $2$ | $ 5 \cdot 41 $ | 4.0.8405.1 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
4.840500.12t34.a.a | $4$ | $ 2^{2} \cdot 5^{3} \cdot 41^{2}$ | 6.2.1378420.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.27568400.12t34.b.a | $4$ | $ 2^{4} \cdot 5^{2} \cdot 41^{3}$ | 6.2.1378420.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.33620.6t13.a.a | $4$ | $ 2^{2} \cdot 5 \cdot 41^{2}$ | 6.2.1378420.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.16400.6t13.b.a | $4$ | $ 2^{4} \cdot 5^{2} \cdot 41 $ | 6.2.1378420.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
6.53792000.12t201.a.a | $6$ | $ 2^{8} \cdot 5^{3} \cdot 41^{2}$ | 8.4.11027360000.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
6.2205472000.12t202.a.a | $6$ | $ 2^{8} \cdot 5^{3} \cdot 41^{3}$ | 8.4.11027360000.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
* | 6.2205472000.8t47.a.a | $6$ | $ 2^{8} \cdot 5^{3} \cdot 41^{3}$ | 8.4.11027360000.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ |
6.90424352000.12t200.a.a | $6$ | $ 2^{8} \cdot 5^{3} \cdot 41^{4}$ | 8.4.11027360000.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.593...000.16t1294.a.a | $9$ | $ 2^{12} \cdot 5^{3} \cdot 41^{5}$ | 8.4.11027360000.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.144...000.18t272.a.a | $9$ | $ 2^{12} \cdot 5^{3} \cdot 41^{4}$ | 8.4.11027360000.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.180...000.18t273.a.a | $9$ | $ 2^{12} \cdot 5^{6} \cdot 41^{4}$ | 8.4.11027360000.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.741...000.18t274.a.a | $9$ | $ 2^{12} \cdot 5^{6} \cdot 41^{5}$ | 8.4.11027360000.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
12.199...000.36t1763.a.a | $12$ | $ 2^{16} \cdot 5^{6} \cdot 41^{7}$ | 8.4.11027360000.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
12.118...000.24t2821.a.a | $12$ | $ 2^{16} \cdot 5^{6} \cdot 41^{5}$ | 8.4.11027360000.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
18.107...000.36t1758.a.a | $18$ | $ 2^{24} \cdot 5^{9} \cdot 41^{9}$ | 8.4.11027360000.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ |