Normalized defining polynomial
\( x^{8} - 16x^{6} - 16x^{5} + 78x^{4} + 128x^{3} - 48x^{2} - 112x + 20 \)
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: | \(14832757637120\) \(\medspace = 2^{22}\cdot 5\cdot 29^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(44.30\) | 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: | $2^{11/4}5^{1/2}29^{1/2}\approx 81.00586972196002$ | ||
Ramified primes: | \(2\), \(5\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{46}a^{7}+\frac{2}{23}a^{6}+\frac{7}{46}a^{4}+\frac{7}{23}a^{3}-\frac{1}{23}a+\frac{9}{23}$
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: | $\frac{5}{46}a^{7}-\frac{3}{46}a^{6}-\frac{3}{2}a^{5}-\frac{17}{23}a^{4}+\frac{150}{23}a^{3}+7a^{2}-\frac{51}{23}a-\frac{1}{23}$, $\frac{1}{2}a^{4}-4a^{2}-4a+1$, $\frac{4}{23}a^{7}+\frac{9}{46}a^{6}-\frac{3}{2}a^{5}-\frac{64}{23}a^{4}+\frac{33}{23}a^{3}+3a^{2}-\frac{31}{23}a+\frac{3}{23}$, $\frac{6}{23}a^{7}+\frac{1}{23}a^{6}-2a^{5}-\frac{50}{23}a^{4}+\frac{15}{23}a^{3}+2a^{2}+\frac{34}{23}a-\frac{7}{23}$, $\frac{79}{46}a^{7}+\frac{201}{46}a^{6}-\frac{27}{2}a^{5}-\frac{2483}{46}a^{4}-\frac{436}{23}a^{3}+93a^{2}+\frac{1554}{23}a-\frac{577}{23}$ | 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: | \( 11472.7121883 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 11472.7121883 \cdot 1}{2\cdot\sqrt{14832757637120}}\cr\approx \mathstrut & 0.940816384616 \end{aligned}\]
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
\(\Q(\sqrt{29}) \) |
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 | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.22.34 | $x^{8} + 16 x^{7} + 72 x^{6} + 80 x^{5} + 188 x^{4} + 32 x^{3} + 336 x^{2} - 32 x + 436$ | $4$ | $2$ | $22$ | $C_8:C_2$ | $[3, 4]^{4}$ |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(29\) | 29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
29.6.3.1 | $x^{6} + 2088 x^{5} + 1453339 x^{4} + 337280262 x^{3} + 45109591 x^{2} + 715298970 x + 9129666150$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.145.2t1.a.a | $1$ | $ 5 \cdot 29 $ | \(\Q(\sqrt{145}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.29.2t1.a.a | $1$ | $ 29 $ | \(\Q(\sqrt{29}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.9280.4t3.p.a | $2$ | $ 2^{6} \cdot 5 \cdot 29 $ | 4.0.46400.1 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
4.46400.6t13.a.a | $4$ | $ 2^{6} \cdot 5^{2} \cdot 29 $ | 6.2.232000.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.6728000.12t34.c.a | $4$ | $ 2^{6} \cdot 5^{3} \cdot 29^{2}$ | 6.2.232000.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.39022400.12t34.b.a | $4$ | $ 2^{6} \cdot 5^{2} \cdot 29^{3}$ | 6.2.232000.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.269120.6t13.a.a | $4$ | $ 2^{6} \cdot 5 \cdot 29^{2}$ | 6.2.232000.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
6.998973440000.12t201.a.a | $6$ | $ 2^{16} \cdot 5^{4} \cdot 29^{3}$ | 8.4.14832757637120.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
6.319...000.12t202.a.a | $6$ | $ 2^{22} \cdot 5^{5} \cdot 29^{3}$ | 8.4.14832757637120.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
* | 6.511474401280.8t47.a.a | $6$ | $ 2^{22} \cdot 5 \cdot 29^{3}$ | 8.4.14832757637120.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ |
6.39958937600.12t200.a.a | $6$ | $ 2^{16} \cdot 5^{2} \cdot 29^{3}$ | 8.4.14832757637120.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.127...000.16t1294.b.a | $9$ | $ 2^{22} \cdot 5^{3} \cdot 29^{3}$ | 8.4.14832757637120.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.159...000.18t272.a.a | $9$ | $ 2^{22} \cdot 5^{6} \cdot 29^{3}$ | 8.4.14832757637120.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.389...000.18t273.b.a | $9$ | $ 2^{22} \cdot 5^{6} \cdot 29^{6}$ | 8.4.14832757637120.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.311...000.18t274.a.a | $9$ | $ 2^{22} \cdot 5^{3} \cdot 29^{6}$ | 8.4.14832757637120.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
12.127...000.36t1763.a.a | $12$ | $ 2^{38} \cdot 5^{7} \cdot 29^{6}$ | 8.4.14832757637120.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
12.510...000.24t2821.a.a | $12$ | $ 2^{38} \cdot 5^{5} \cdot 29^{6}$ | 8.4.14832757637120.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
18.319...000.36t1758.b.a | $18$ | $ 2^{50} \cdot 5^{9} \cdot 29^{9}$ | 8.4.14832757637120.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ |