Normalized defining polynomial
\( x^{7} - 7x^{5} + 13x^{3} - 5x - 1 \)
Invariants
Degree: | $7$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[7, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(49960857\) \(\medspace = 3\cdot 16653619\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.58\) | 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: | $3^{1/2}16653619^{1/2}\approx 7068.299441874262$ | ||
Ramified primes: | \(3\), \(16653619\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{49960857}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $6$ | 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: | $a$, $a^{6}-6a^{4}+8a^{2}-1$, $a^{6}-7a^{4}+a^{3}+12a^{2}-4a-2$, $a^{6}-7a^{4}+12a^{2}-2$, $a^{6}-7a^{4}+12a^{2}-a-2$, $a^{6}-7a^{4}+13a^{2}-a-4$ | 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: | \( 30.5453900289 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 30.5453900289 \cdot 1}{2\cdot\sqrt{49960857}}\cr\approx \mathstrut & 0.276573591417 \end{aligned}\]
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, deg 42 |
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 | ${\href{/padicField/2.7.0.1}{7} }$ | R | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.5.0.1 | $x^{5} + 2 x + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
\(16653619\) | $\Q_{16653619}$ | $x$ | $1$ | $1$ | $0$ | 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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.49960857.2t1.a.a | $1$ | $ 3 \cdot 16653619 $ | \(\Q(\sqrt{49960857}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
6.311...057.14t46.a.a | $6$ | $ 3^{5} \cdot 16653619^{5}$ | 7.7.49960857.1 | $S_7$ (as 7T7) | $1$ | $6$ | |
* | 6.49960857.7t7.a.a | $6$ | $ 3 \cdot 16653619 $ | 7.7.49960857.1 | $S_7$ (as 7T7) | $1$ | $6$ |
14.623...601.21t38.a.a | $14$ | $ 3^{4} \cdot 16653619^{4}$ | 7.7.49960857.1 | $S_7$ (as 7T7) | $1$ | $14$ | |
14.968...249.42t413.a.a | $14$ | $ 3^{10} \cdot 16653619^{10}$ | 7.7.49960857.1 | $S_7$ (as 7T7) | $1$ | $14$ | |
14.193...257.30t565.a.a | $14$ | $ 3^{9} \cdot 16653619^{9}$ | 7.7.49960857.1 | $S_7$ (as 7T7) | $1$ | $14$ | |
14.311...057.30t565.a.a | $14$ | $ 3^{5} \cdot 16653619^{5}$ | 7.7.49960857.1 | $S_7$ (as 7T7) | $1$ | $14$ | |
15.311...057.42t412.a.a | $15$ | $ 3^{5} \cdot 16653619^{5}$ | 7.7.49960857.1 | $S_7$ (as 7T7) | $1$ | $15$ | |
15.968...249.42t411.a.a | $15$ | $ 3^{10} \cdot 16653619^{10}$ | 7.7.49960857.1 | $S_7$ (as 7T7) | $1$ | $15$ | |
20.968...249.70.a.a | $20$ | $ 3^{10} \cdot 16653619^{10}$ | 7.7.49960857.1 | $S_7$ (as 7T7) | $1$ | $20$ | |
21.968...249.84.a.a | $21$ | $ 3^{10} \cdot 16653619^{10}$ | 7.7.49960857.1 | $S_7$ (as 7T7) | $1$ | $21$ | |
21.484...393.42t418.a.a | $21$ | $ 3^{11} \cdot 16653619^{11}$ | 7.7.49960857.1 | $S_7$ (as 7T7) | $1$ | $21$ | |
35.938...001.126.a.a | $35$ | $ 3^{20} \cdot 16653619^{20}$ | 7.7.49960857.1 | $S_7$ (as 7T7) | $1$ | $35$ | |
35.301...193.70.a.a | $35$ | $ 3^{15} \cdot 16653619^{15}$ | 7.7.49960857.1 | $S_7$ (as 7T7) | $1$ | $35$ |