Normalized defining polynomial
\( x^{8} - 4x^{7} - 21x^{4} - 21x^{2} - 15x - 3 \)
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: | \(-375226779375\) \(\medspace = -\,3^{6}\cdot 5^{4}\cdot 7^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(27.98\) | 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: | $3^{6/7}5^{2/3}7^{47/42}\approx 66.16769834619991$ | ||
Ramified primes: | \(3\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a^{2}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{6}+\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a^{2}-\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{25}a^{7}-\frac{1}{25}a^{6}+\frac{2}{25}a^{5}-\frac{4}{25}a^{4}+\frac{12}{25}a^{3}-\frac{4}{25}a^{2}-\frac{8}{25}a-\frac{9}{25}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{3}{25}a^{7}-\frac{18}{25}a^{6}+\frac{26}{25}a^{5}-\frac{7}{25}a^{4}-\frac{64}{25}a^{3}+\frac{108}{25}a^{2}-\frac{79}{25}a+\frac{53}{25}$, $\frac{14}{25}a^{7}-\frac{64}{25}a^{6}+\frac{33}{25}a^{5}-\frac{1}{25}a^{4}-\frac{297}{25}a^{3}+\frac{134}{25}a^{2}-\frac{342}{25}a-\frac{61}{25}$, $\frac{134}{25}a^{7}-\frac{569}{25}a^{6}+\frac{138}{25}a^{5}-\frac{26}{25}a^{4}-\frac{2812}{25}a^{3}+\frac{714}{25}a^{2}-\frac{2932}{25}a-\frac{1291}{25}$, $\frac{276}{25}a^{7}-\frac{1176}{25}a^{6}+\frac{302}{25}a^{5}-\frac{54}{25}a^{4}-\frac{5788}{25}a^{3}+\frac{1521}{25}a^{2}-\frac{6083}{25}a-\frac{2584}{25}$ | 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: | \( 1356.29591837 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 1356.29591837 \cdot 1}{2\cdot\sqrt{375226779375}}\cr\approx \mathstrut & 1.09844203144 \end{aligned}\]
Galois group
$\PGL(2,7)$ (as 8T43):
A non-solvable group of order 336 |
The 9 conjugacy class representatives for $\PGL(2,7)$ |
Character table for $\PGL(2,7)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 14 sibling: | deg 14 |
Degree 16 sibling: | deg 16 |
Degree 21 sibling: | deg 21 |
Degree 24 sibling: | deg 24 |
Degree 28 siblings: | deg 28, deg 28 |
Degree 42 siblings: | deg 42, deg 42, some 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 | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | R | R | R | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.7.6.1 | $x^{7} + 3$ | $7$ | $1$ | $6$ | $F_7$ | $[\ ]_{7}^{6}$ | |
\(5\) | 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.7.7.6 | $x^{7} + 28 x + 7$ | $7$ | $1$ | $7$ | $F_7$ | $[7/6]_{6}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
6.375226779375.42t82.a.a | $6$ | $ 3^{6} \cdot 5^{4} \cdot 7^{7}$ | 8.2.375226779375.1 | $\PGL(2,7)$ (as 8T43) | $1$ | $0$ | |
6.375226779375.14t16.a.a | $6$ | $ 3^{6} \cdot 5^{4} \cdot 7^{7}$ | 8.2.375226779375.1 | $\PGL(2,7)$ (as 8T43) | $1$ | $0$ | |
6.375226779375.14t16.a.b | $6$ | $ 3^{6} \cdot 5^{4} \cdot 7^{7}$ | 8.2.375226779375.1 | $\PGL(2,7)$ (as 8T43) | $1$ | $0$ | |
* | 7.375226779375.8t43.a.a | $7$ | $ 3^{6} \cdot 5^{4} \cdot 7^{7}$ | 8.2.375226779375.1 | $\PGL(2,7)$ (as 8T43) | $1$ | $1$ |
7.262...625.16t713.a.a | $7$ | $ 3^{6} \cdot 5^{4} \cdot 7^{8}$ | 8.2.375226779375.1 | $\PGL(2,7)$ (as 8T43) | $1$ | $-1$ | |
8.459...375.42t81.a.a | $8$ | $ 3^{6} \cdot 5^{6} \cdot 7^{9}$ | 8.2.375226779375.1 | $\PGL(2,7)$ (as 8T43) | $1$ | $-2$ | |
8.459...375.21t20.a.a | $8$ | $ 3^{6} \cdot 5^{6} \cdot 7^{9}$ | 8.2.375226779375.1 | $\PGL(2,7)$ (as 8T43) | $1$ | $2$ |