Normalized defining polynomial
\( x^{8} - 3x^{7} - 7x^{6} + 35x^{5} - 21x^{4} - 77x^{3} + 189x^{2} - 171x + 72 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(2420662999104\) \(\medspace = 2^{6}\cdot 3^{8}\cdot 7^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(35.32\) | 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: | $2^{6/7}3^{4/3}7^{26/21}\approx 87.19653805851985$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{21}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{7}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $3$ | 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: | $\frac{1013}{21}a^{7}+32a^{6}-\frac{1988}{3}a^{5}-\frac{206}{3}a^{4}+3331a^{3}-\frac{2482}{3}a^{2}-4142a+\frac{42457}{7}$, $\frac{2}{21}a^{7}-8a^{6}+\frac{70}{3}a^{5}-\frac{23}{3}a^{4}-63a^{3}+\frac{407}{3}a^{2}-119a+\frac{341}{7}$, $\frac{11758}{21}a^{7}-3099a^{6}-\frac{7450}{3}a^{5}+\frac{106487}{3}a^{4}-34484a^{3}-\frac{273272}{3}a^{2}+185310a-\frac{928093}{7}$ (assuming GRH) | 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: | \( 13681.3207265 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 13681.3207265 \cdot 1}{2\cdot\sqrt{2420662999104}}\cr\approx \mathstrut & 6.85252038046 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 168 |
The 8 conjugacy class representatives for $C_2^3:(C_7: C_3)$ |
Character table for $C_2^3:(C_7: C_3)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 14 sibling: | deg 14 |
Degree 24 sibling: | deg 24 |
Degree 28 sibling: | deg 28 |
Degree 42 sibling: | 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 | ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
3.3.4.3 | $x^{3} + 6 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
3.3.4.3 | $x^{3} + 6 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.7.8.1 | $x^{7} + 14 x^{2} + 7$ | $7$ | $1$ | $8$ | $C_7:C_3$ | $[4/3]_{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.63.3t1.a.a | $1$ | $ 3^{2} \cdot 7 $ | 3.3.3969.2 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.63.3t1.a.b | $1$ | $ 3^{2} \cdot 7 $ | 3.3.3969.2 | $C_3$ (as 3T1) | $0$ | $1$ | |
3.1555848.7t3.a.a | $3$ | $ 2^{3} \cdot 3^{4} \cdot 7^{4}$ | 7.7.2420662999104.1 | $C_7:C_3$ (as 7T3) | $0$ | $3$ | |
3.1555848.7t3.a.b | $3$ | $ 2^{3} \cdot 3^{4} \cdot 7^{4}$ | 7.7.2420662999104.1 | $C_7:C_3$ (as 7T3) | $0$ | $3$ | |
* | 7.242...104.8t36.a.a | $7$ | $ 2^{6} \cdot 3^{8} \cdot 7^{8}$ | 8.0.2420662999104.2 | $C_2^3:(C_7: C_3)$ (as 8T36) | $1$ | $-1$ |
7.152...552.24t283.a.a | $7$ | $ 2^{6} \cdot 3^{10} \cdot 7^{9}$ | 8.0.2420662999104.2 | $C_2^3:(C_7: C_3)$ (as 8T36) | $0$ | $-1$ | |
7.152...552.24t283.a.b | $7$ | $ 2^{6} \cdot 3^{10} \cdot 7^{9}$ | 8.0.2420662999104.2 | $C_2^3:(C_7: C_3)$ (as 8T36) | $0$ | $-1$ |