Normalized defining polynomial
\( x^{7} - x^{5} - x^{4} - x^{3} + x^{2} + x + 1 \)
Invariants
Degree: | $7$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-207911\) \(\medspace = -\,11\cdot 41\cdot 461\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(5.75\) | 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: | $11^{1/2}41^{1/2}461^{1/2}\approx 455.9725868953089$ | ||
Ramified primes: | \(11\), \(41\), \(461\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-207911}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $a$, $a^{6}-a^{4}-a^{2}$, $a^{5}-a^{3}$ | 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: | \( 0.413665504587 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{3}\cdot 0.413665504587 \cdot 1}{2\cdot\sqrt{207911}}\cr\approx \mathstrut & 0.2250350561757 \end{aligned}\]
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, 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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | R | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | R | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.5.0.1 | $x^{5} + 10 x^{2} + 9$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
\(41\) | $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
41.2.1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
41.4.0.1 | $x^{4} + 23 x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(461\) | $\Q_{461}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{461}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.207911.2t1.a.a | $1$ | $ 11 \cdot 41 \cdot 461 $ | \(\Q(\sqrt{-207911}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
6.388...551.14t46.a.a | $6$ | $ 11^{5} \cdot 41^{5} \cdot 461^{5}$ | 7.1.207911.1 | $S_7$ (as 7T7) | $1$ | $0$ | |
* | 6.207911.7t7.a.a | $6$ | $ 11 \cdot 41 \cdot 461 $ | 7.1.207911.1 | $S_7$ (as 7T7) | $1$ | $0$ |
14.186...241.21t38.a.a | $14$ | $ 11^{4} \cdot 41^{4} \cdot 461^{4}$ | 7.1.207911.1 | $S_7$ (as 7T7) | $1$ | $2$ | |
14.150...601.42t413.a.a | $14$ | $ 11^{10} \cdot 41^{10} \cdot 461^{10}$ | 7.1.207911.1 | $S_7$ (as 7T7) | $1$ | $-2$ | |
14.725...791.30t565.a.a | $14$ | $ 11^{9} \cdot 41^{9} \cdot 461^{9}$ | 7.1.207911.1 | $S_7$ (as 7T7) | $1$ | $0$ | |
14.388...551.30t565.a.a | $14$ | $ 11^{5} \cdot 41^{5} \cdot 461^{5}$ | 7.1.207911.1 | $S_7$ (as 7T7) | $1$ | $0$ | |
15.388...551.42t412.a.a | $15$ | $ 11^{5} \cdot 41^{5} \cdot 461^{5}$ | 7.1.207911.1 | $S_7$ (as 7T7) | $1$ | $-3$ | |
15.150...601.42t411.a.a | $15$ | $ 11^{10} \cdot 41^{10} \cdot 461^{10}$ | 7.1.207911.1 | $S_7$ (as 7T7) | $1$ | $3$ | |
20.150...601.70.a.a | $20$ | $ 11^{10} \cdot 41^{10} \cdot 461^{10}$ | 7.1.207911.1 | $S_7$ (as 7T7) | $1$ | $0$ | |
21.150...601.84.a.a | $21$ | $ 11^{10} \cdot 41^{10} \cdot 461^{10}$ | 7.1.207911.1 | $S_7$ (as 7T7) | $1$ | $-3$ | |
21.313...511.42t418.a.a | $21$ | $ 11^{11} \cdot 41^{11} \cdot 461^{11}$ | 7.1.207911.1 | $S_7$ (as 7T7) | $1$ | $3$ | |
35.227...201.126.a.a | $35$ | $ 11^{20} \cdot 41^{20} \cdot 461^{20}$ | 7.1.207911.1 | $S_7$ (as 7T7) | $1$ | $-1$ | |
35.586...151.70.a.a | $35$ | $ 11^{15} \cdot 41^{15} \cdot 461^{15}$ | 7.1.207911.1 | $S_7$ (as 7T7) | $1$ | $1$ |