Normalized defining polynomial
\( x^{7} - x^{5} - 3x^{4} - 3x^{3} + 5x^{2} + 3x - 1 \)
Invariants
Degree: | $7$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[5, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(-5280527\)
\(\medspace = -\,7\cdot 353\cdot 2137\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(9.13\) | 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: | $7^{1/2}353^{1/2}2137^{1/2}\approx 2297.9397294098035$ | ||
Ramified primes: |
\(7\), \(353\), \(2137\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-5280527}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}+\frac{2}{7}a^{5}+\frac{3}{7}a^{4}+\frac{3}{7}a^{3}+\frac{3}{7}a^{2}-\frac{3}{7}a-\frac{3}{7}$
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: |
$\frac{4}{7}a^{6}+\frac{1}{7}a^{5}-\frac{2}{7}a^{4}-\frac{9}{7}a^{3}-\frac{16}{7}a^{2}+\frac{16}{7}a+\frac{2}{7}$, $a$, $\frac{2}{7}a^{6}-\frac{3}{7}a^{5}-\frac{1}{7}a^{4}-\frac{8}{7}a^{3}-\frac{1}{7}a^{2}+\frac{15}{7}a+\frac{1}{7}$, $\frac{6}{7}a^{6}-\frac{2}{7}a^{5}-\frac{3}{7}a^{4}-\frac{17}{7}a^{3}-\frac{10}{7}a^{2}+\frac{24}{7}a+\frac{3}{7}$, $\frac{6}{7}a^{6}-\frac{2}{7}a^{5}-\frac{3}{7}a^{4}-\frac{17}{7}a^{3}-\frac{10}{7}a^{2}+\frac{31}{7}a+\frac{3}{7}$
| 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: | \( 5.01412562612 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{1}\cdot 5.01412562612 \cdot 1}{2\cdot\sqrt{5280527}}\cr\approx \mathstrut & 0.219359490132 \end{aligned}\]
Galois group
A non-solvable group of order 5040 |
The 15 conjugacy class representatives for $S_7$ |
Character table for $S_7$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 14 sibling: | deg 14 |
Degree 21 sibling: | deg 21 |
Degree 30 sibling: | deg 30 |
Degree 35 sibling: | deg 35 |
Degree 42 siblings: | 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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(353\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
\(2137\)
| $\Q_{2137}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.5280527.2t1.a.a | $1$ | $ 7 \cdot 353 \cdot 2137 $ | \(\Q(\sqrt{-5280527}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
6.410...407.14t46.a.a | $6$ | $ 7^{5} \cdot 353^{5} \cdot 2137^{5}$ | 7.5.5280527.1 | $S_7$ (as 7T7) | $1$ | $-4$ | |
* | 6.5280527.7t7.a.a | $6$ | $ 7 \cdot 353 \cdot 2137 $ | 7.5.5280527.1 | $S_7$ (as 7T7) | $1$ | $4$ |
14.777...441.21t38.a.a | $14$ | $ 7^{4} \cdot 353^{4} \cdot 2137^{4}$ | 7.5.5280527.1 | $S_7$ (as 7T7) | $1$ | $6$ | |
14.168...649.42t413.a.a | $14$ | $ 7^{10} \cdot 353^{10} \cdot 2137^{10}$ | 7.5.5280527.1 | $S_7$ (as 7T7) | $1$ | $-6$ | |
14.319...487.30t565.a.a | $14$ | $ 7^{9} \cdot 353^{9} \cdot 2137^{9}$ | 7.5.5280527.1 | $S_7$ (as 7T7) | $1$ | $-4$ | |
14.410...407.30t565.a.a | $14$ | $ 7^{5} \cdot 353^{5} \cdot 2137^{5}$ | 7.5.5280527.1 | $S_7$ (as 7T7) | $1$ | $4$ | |
15.410...407.42t412.a.a | $15$ | $ 7^{5} \cdot 353^{5} \cdot 2137^{5}$ | 7.5.5280527.1 | $S_7$ (as 7T7) | $1$ | $5$ | |
15.168...649.42t411.a.a | $15$ | $ 7^{10} \cdot 353^{10} \cdot 2137^{10}$ | 7.5.5280527.1 | $S_7$ (as 7T7) | $1$ | $-5$ | |
20.168...649.70.a.a | $20$ | $ 7^{10} \cdot 353^{10} \cdot 2137^{10}$ | 7.5.5280527.1 | $S_7$ (as 7T7) | $1$ | $0$ | |
21.168...649.84.a.a | $21$ | $ 7^{10} \cdot 353^{10} \cdot 2137^{10}$ | 7.5.5280527.1 | $S_7$ (as 7T7) | $1$ | $1$ | |
21.890...023.42t418.a.a | $21$ | $ 7^{11} \cdot 353^{11} \cdot 2137^{11}$ | 7.5.5280527.1 | $S_7$ (as 7T7) | $1$ | $-1$ | |
35.284...201.126.a.a | $35$ | $ 7^{20} \cdot 353^{20} \cdot 2137^{20}$ | 7.5.5280527.1 | $S_7$ (as 7T7) | $1$ | $-5$ | |
35.692...143.70.a.a | $35$ | $ 7^{15} \cdot 353^{15} \cdot 2137^{15}$ | 7.5.5280527.1 | $S_7$ (as 7T7) | $1$ | $5$ |