Normalized defining polynomial
\( x^{8} + 4x^{6} - 4x^{5} - 18x^{4} - 8x^{3} - 40x^{2} + 44x - 15 \)
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: | \(-82104483840\) \(\medspace = -\,2^{16}\cdot 3\cdot 5\cdot 17^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(23.14\) | 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: | $2^{25/12}3^{1/2}5^{1/2}17^{1/2}\approx 67.67307556999491$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-15}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{1740}a^{7}+\frac{169}{1740}a^{6}-\frac{1}{12}a^{5}-\frac{149}{1740}a^{4}-\frac{839}{1740}a^{3}-\frac{859}{1740}a^{2}+\frac{79}{1740}a-\frac{35}{116}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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: | $\frac{3}{145}a^{7}-\frac{1}{290}a^{6}-\frac{12}{145}a^{4}-\frac{52}{145}a^{3}-\frac{79}{290}a^{2}-\frac{53}{145}a+\frac{4}{29}$, $\frac{1}{15}a^{7}+\frac{1}{60}a^{6}+\frac{1}{3}a^{5}-\frac{11}{60}a^{4}-\frac{43}{30}a^{3}-\frac{61}{60}a^{2}-\frac{97}{30}a+\frac{9}{4}$, $\frac{14}{435}a^{7}+\frac{329}{1740}a^{6}-\frac{1}{6}a^{5}+\frac{791}{1740}a^{4}-\frac{1307}{870}a^{3}-\frac{5909}{1740}a^{2}+\frac{2411}{435}a-\frac{249}{116}$, $\frac{14}{145}a^{7}-\frac{251}{580}a^{6}+\frac{1}{2}a^{5}-\frac{659}{580}a^{4}-\frac{147}{290}a^{3}+\frac{5111}{580}a^{2}-\frac{1069}{145}a+\frac{239}{116}$ | 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: | \( 519.058800592 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 519.058800592 \cdot 2}{2\cdot\sqrt{82104483840}}\cr\approx \mathstrut & 1.79735025522 \end{aligned}\]
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
\(\Q(\sqrt{34}) \) |
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 | R | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.8.0.1}{8} }$ | R | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.16.58 | $x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{2} + 4 x + 2$ | $8$ | $1$ | $16$ | $S_4\times C_2$ | $[4/3, 4/3, 3]_{3}^{2}$ |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(17\) | 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
17.6.3.1 | $x^{6} + 459 x^{5} + 70280 x^{4} + 3597823 x^{3} + 1271380 x^{2} + 4696159 x + 50437479$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.2040.2t1.b.a | $1$ | $ 2^{3} \cdot 3 \cdot 5 \cdot 17 $ | \(\Q(\sqrt{-510}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.136.2t1.a.a | $1$ | $ 2^{3} \cdot 17 $ | \(\Q(\sqrt{34}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.2040.4t3.l.a | $2$ | $ 2^{3} \cdot 3 \cdot 5 \cdot 17 $ | 4.0.30600.4 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.122400.6t13.b.a | $4$ | $ 2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 17 $ | 6.0.1836000.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.249696000.12t34.h.a | $4$ | $ 2^{8} \cdot 3^{3} \cdot 5^{3} \cdot 17^{2}$ | 6.0.1836000.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.565977600.12t34.b.a | $4$ | $ 2^{9} \cdot 3^{2} \cdot 5^{2} \cdot 17^{3}$ | 6.0.1836000.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.1109760.6t13.b.a | $4$ | $ 2^{8} \cdot 3 \cdot 5 \cdot 17^{2}$ | 6.0.1836000.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.203...000.12t201.a.a | $6$ | $ 2^{13} \cdot 3^{4} \cdot 5^{4} \cdot 17^{3}$ | 8.2.82104483840.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
6.305...000.12t202.a.a | $6$ | $ 2^{13} \cdot 3^{5} \cdot 5^{5} \cdot 17^{3}$ | 8.2.82104483840.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
* | 6.603709440.8t47.a.a | $6$ | $ 2^{13} \cdot 3 \cdot 5 \cdot 17^{3}$ | 8.2.82104483840.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
6.9055641600.12t200.a.a | $6$ | $ 2^{13} \cdot 3^{2} \cdot 5^{2} \cdot 17^{3}$ | 8.2.82104483840.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.543338496000.16t1294.a.a | $9$ | $ 2^{15} \cdot 3^{3} \cdot 5^{3} \cdot 17^{3}$ | 8.2.82104483840.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
9.183...000.18t272.a.a | $9$ | $ 2^{15} \cdot 3^{6} \cdot 5^{6} \cdot 17^{3}$ | 8.2.82104483840.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.115...000.18t273.a.a | $9$ | $ 2^{22} \cdot 3^{6} \cdot 5^{6} \cdot 17^{6}$ | 8.2.82104483840.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.341...000.18t274.a.a | $9$ | $ 2^{22} \cdot 3^{3} \cdot 5^{3} \cdot 17^{6}$ | 8.2.82104483840.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
12.276...000.36t1763.a.a | $12$ | $ 2^{26} \cdot 3^{7} \cdot 5^{7} \cdot 17^{6}$ | 8.2.82104483840.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.123...000.24t2821.a.a | $12$ | $ 2^{26} \cdot 3^{5} \cdot 5^{5} \cdot 17^{6}$ | 8.2.82104483840.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.626...000.36t1758.a.a | $18$ | $ 2^{37} \cdot 3^{9} \cdot 5^{9} \cdot 17^{9}$ | 8.2.82104483840.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |