Normalized defining polynomial
\( x^{8} - 2x^{7} - 9x^{6} + 32x^{5} + 18x^{4} - 125x^{3} + 46x^{2} + 165x - 94 \)
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: | \(1043729299208\) \(\medspace = 2^{3}\cdot 601^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(31.79\) | 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: | $2^{3/2}601^{1/2}\approx 69.33974329343886$ | ||
Ramified primes: | \(2\), \(601\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{6946}a^{7}+\frac{1397}{6946}a^{6}+\frac{1284}{3473}a^{5}+\frac{791}{3473}a^{4}-\frac{1269}{3473}a^{3}-\frac{1381}{6946}a^{2}-\frac{985}{6946}a-\frac{1271}{3473}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | $\frac{221}{3473}a^{7}-\frac{360}{3473}a^{6}-\frac{2044}{3473}a^{5}+\frac{5795}{3473}a^{4}+\frac{5201}{3473}a^{3}-\frac{20415}{3473}a^{2}+\frac{1114}{3473}a+\frac{25155}{3473}$, $\frac{362}{3473}a^{7}-\frac{1344}{3473}a^{6}-\frac{1148}{3473}a^{5}+\frac{13531}{3473}a^{4}-\frac{15776}{3473}a^{3}-\frac{20648}{3473}a^{2}+\frac{49771}{3473}a-\frac{27643}{3473}$, $\frac{112}{3473}a^{7}+\frac{179}{3473}a^{6}-\frac{643}{3473}a^{5}+\frac{61}{3473}a^{4}+\frac{4003}{3473}a^{3}+\frac{5086}{3473}a^{2}+\frac{816}{3473}a-\frac{3391}{3473}$, $\frac{122}{3473}a^{7}+\frac{257}{3473}a^{6}-\frac{2747}{3473}a^{5}+\frac{1989}{3473}a^{4}+\frac{20299}{3473}a^{3}-\frac{29562}{3473}a^{2}-\frac{29872}{3473}a+\frac{68433}{3473}$, $\frac{207824}{3473}a^{7}-\frac{370391}{3473}a^{6}-\frac{1233320}{3473}a^{5}+\frac{4423679}{3473}a^{4}-\frac{1961155}{3473}a^{3}-\frac{7991070}{3473}a^{2}+\frac{12345441}{3473}a-\frac{4605357}{3473}$ | 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: | \( 10109.8797349 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 10109.8797349 \cdot 1}{2\cdot\sqrt{1043729299208}}\cr\approx \mathstrut & 3.12537242121 \end{aligned}\]
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
\(\Q(\sqrt{601}) \) |
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 | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(601\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $6$ | $2$ | $3$ | $3$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.4808.2t1.a.a | $1$ | $ 2^{3} \cdot 601 $ | \(\Q(\sqrt{1202}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.601.2t1.a.a | $1$ | $ 601 $ | \(\Q(\sqrt{601}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.4808.4t3.e.a | $2$ | $ 2^{3} \cdot 601 $ | 4.0.38464.2 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
4.13893235264.12t34.d.a | $4$ | $ 2^{6} \cdot 601^{3}$ | 6.2.307712.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.184934912.12t34.b.a | $4$ | $ 2^{9} \cdot 601^{2}$ | 6.2.307712.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.38464.6t13.d.a | $4$ | $ 2^{6} \cdot 601 $ | 6.2.307712.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.2889608.6t13.b.a | $4$ | $ 2^{3} \cdot 601^{2}$ | 6.2.307712.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
6.889167056896.12t201.a.a | $6$ | $ 2^{12} \cdot 601^{3}$ | 8.4.1043729299208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
6.711...168.12t202.a.a | $6$ | $ 2^{15} \cdot 601^{3}$ | 8.4.1043729299208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
* | 6.1736654408.8t47.a.a | $6$ | $ 2^{3} \cdot 601^{3}$ | 8.4.1043729299208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ |
6.13893235264.12t200.a.a | $6$ | $ 2^{6} \cdot 601^{3}$ | 8.4.1043729299208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.111145882112.16t1294.a.a | $9$ | $ 2^{9} \cdot 601^{3}$ | 8.4.1043729299208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.569...344.18t272.a.a | $9$ | $ 2^{18} \cdot 601^{3}$ | 8.4.1043729299208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.123...544.18t273.a.a | $9$ | $ 2^{18} \cdot 601^{6}$ | 8.4.1043729299208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.241...712.18t274.a.a | $9$ | $ 2^{9} \cdot 601^{6}$ | 8.4.1043729299208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
12.988...352.36t1763.a.a | $12$ | $ 2^{21} \cdot 601^{6}$ | 8.4.1043729299208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
12.154...568.24t2821.a.a | $12$ | $ 2^{15} \cdot 601^{6}$ | 8.4.1043729299208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
18.137...928.36t1758.a.a | $18$ | $ 2^{27} \cdot 601^{9}$ | 8.4.1043729299208.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ |