Normalized defining polynomial
\( x^{8} - 23x^{6} - 12x^{5} + 229x^{4} + 73x^{3} - 971x^{2} + 37x + 1226 \)
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: | \(28298328053125\) \(\medspace = 5^{5}\cdot 13^{5}\cdot 29^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(48.03\) | 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: | $5^{3/4}13^{3/4}29^{1/2}\approx 123.27755617119755$ | ||
Ramified primes: | \(5\), \(13\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{1885}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{45}a^{6}-\frac{17}{45}a^{5}-\frac{1}{15}a^{4}+\frac{22}{45}a^{3}-\frac{13}{45}a^{2}+\frac{1}{45}a-\frac{11}{45}$, $\frac{1}{3960}a^{7}-\frac{7}{792}a^{6}-\frac{137}{660}a^{5}-\frac{97}{1980}a^{4}+\frac{1211}{3960}a^{3}+\frac{7}{99}a^{2}-\frac{299}{3960}a+\frac{31}{220}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$
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: | $\frac{7}{220}a^{7}+\frac{19}{220}a^{6}-\frac{61}{110}a^{5}-\frac{39}{22}a^{4}+\frac{129}{44}a^{3}+\frac{512}{55}a^{2}-\frac{1389}{220}a+\frac{171}{110}$, $\frac{65}{396}a^{7}+\frac{145}{396}a^{6}-\frac{557}{198}a^{5}-\frac{1619}{198}a^{4}+\frac{2273}{132}a^{3}+\frac{1601}{33}a^{2}-\frac{14243}{396}a-\frac{15371}{198}$, $\frac{49}{1980}a^{7}-\frac{35}{396}a^{6}-\frac{559}{990}a^{5}+\frac{1847}{990}a^{4}+\frac{2033}{660}a^{3}-\frac{112}{11}a^{2}-\frac{5191}{1980}a+\frac{13121}{990}$, $\frac{557}{990}a^{7}+\frac{1669}{990}a^{6}-\frac{3911}{495}a^{5}-\frac{2998}{99}a^{4}+\frac{2551}{66}a^{3}+\frac{25678}{165}a^{2}-\frac{83999}{990}a-\frac{114944}{495}$, $\frac{361}{1980}a^{7}+\frac{521}{1980}a^{6}-\frac{2797}{990}a^{5}-\frac{5251}{990}a^{4}+\frac{9881}{660}a^{3}+\frac{367}{55}a^{2}-\frac{41323}{1980}a+\frac{121421}{990}$ | 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: | \( 34552.6810873 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 34552.6810873 \cdot 2}{2\cdot\sqrt{28298328053125}}\cr\approx \mathstrut & 4.10280695132 \end{aligned}\]
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
\(\Q(\sqrt{65}) \) |
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 | ${\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{3}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.3.4 | $x^{4} + 15$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.4.3.3 | $x^{4} + 26$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(29\) | 29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.4.2.2 | $x^{4} - 696 x^{2} + 1682$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.1885.2t1.a.a | $1$ | $ 5 \cdot 13 \cdot 29 $ | \(\Q(\sqrt{1885}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.29.2t1.a.a | $1$ | $ 29 $ | \(\Q(\sqrt{29}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.65.2t1.a.a | $1$ | $ 5 \cdot 13 $ | \(\Q(\sqrt{65}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.122525.4t3.c.a | $2$ | $ 5^{2} \cdot 13^{2} \cdot 29 $ | 4.0.230959625.2 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
4.230959625.6t13.a.a | $4$ | $ 5^{3} \cdot 13^{3} \cdot 29^{2}$ | 6.2.435358893125.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.6697829125.12t34.a.a | $4$ | $ 5^{3} \cdot 13^{3} \cdot 29^{3}$ | 6.2.435358893125.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.230959625.12t34.b.a | $4$ | $ 5^{3} \cdot 13^{3} \cdot 29^{2}$ | 6.2.435358893125.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.7964125.6t13.a.a | $4$ | $ 5^{3} \cdot 13^{3} \cdot 29 $ | 6.2.435358893125.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
6.975804415625.12t201.a.a | $6$ | $ 5^{5} \cdot 13^{5} \cdot 29^{2}$ | 8.4.28298328053125.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
6.435358893125.12t202.a.a | $6$ | $ 5^{4} \cdot 13^{4} \cdot 29^{3}$ | 8.4.28298328053125.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
* | 6.435358893125.8t47.a.a | $6$ | $ 5^{4} \cdot 13^{4} \cdot 29^{3}$ | 8.4.28298328053125.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ |
6.820...625.12t200.a.a | $6$ | $ 5^{5} \cdot 13^{5} \cdot 29^{4}$ | 8.4.28298328053125.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.154...125.16t1294.a.a | $9$ | $ 5^{6} \cdot 13^{6} \cdot 29^{5}$ | 8.4.28298328053125.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.346...625.18t272.a.a | $9$ | $ 5^{7} \cdot 13^{7} \cdot 29^{4}$ | 8.4.28298328053125.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.533...625.18t273.a.a | $9$ | $ 5^{6} \cdot 13^{6} \cdot 29^{4}$ | 8.4.28298328053125.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.100...125.18t274.a.a | $9$ | $ 5^{7} \cdot 13^{7} \cdot 29^{5}$ | 8.4.28298328053125.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
12.357...125.36t1763.a.a | $12$ | $ 5^{9} \cdot 13^{9} \cdot 29^{7}$ | 8.4.28298328053125.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
12.424...125.24t2821.a.a | $12$ | $ 5^{9} \cdot 13^{9} \cdot 29^{5}$ | 8.4.28298328053125.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
18.348...125.36t1758.a.a | $18$ | $ 5^{14} \cdot 13^{14} \cdot 29^{9}$ | 8.4.28298328053125.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ |