Normalized defining polynomial
\( x^{7} - 7x^{5} - x^{4} + 11x^{3} + 3x^{2} - 3x - 1 \)
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: | \(30653489\) \(\medspace = 1399\cdot 21911\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.74\) | 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: | $1399^{1/2}21911^{1/2}\approx 5536.559310618825$ | ||
Ramified primes: | \(1399\), \(21911\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{30653489}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a^{6}-7a^{4}-a^{3}+11a^{2}+3a-3$, $2a^{6}-a^{5}-14a^{4}+5a^{3}+22a^{2}-5a-5$, $2a^{6}-a^{5}-13a^{4}+5a^{3}+17a^{2}-5a-3$, $a^{6}-7a^{4}-a^{3}+11a^{2}+3a-2$, $3a^{6}-a^{5}-21a^{4}+4a^{3}+33a^{2}-2a-8$, $3a^{6}-a^{5}-21a^{4}+4a^{3}+33a^{2}-2a-9$ | 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: | \( 22.422780934 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 22.422780934 \cdot 1}{2\cdot\sqrt{30653489}}\cr\approx \mathstrut & 0.25919671393 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(1399\) | $\Q_{1399}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1399}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
\(21911\) | $\Q_{21911}$ | $x$ | $1$ | $1$ | $0$ | 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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.30653489.2t1.a.a | $1$ | $ 1399 \cdot 21911 $ | \(\Q(\sqrt{30653489}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
6.270...449.14t46.a.a | $6$ | $ 1399^{5} \cdot 21911^{5}$ | 7.7.30653489.1 | $S_7$ (as 7T7) | $1$ | $6$ | |
* | 6.30653489.7t7.a.a | $6$ | $ 1399 \cdot 21911 $ | 7.7.30653489.1 | $S_7$ (as 7T7) | $1$ | $6$ |
14.882...641.21t38.a.a | $14$ | $ 1399^{4} \cdot 21911^{4}$ | 7.7.30653489.1 | $S_7$ (as 7T7) | $1$ | $14$ | |
14.732...601.42t413.a.a | $14$ | $ 1399^{10} \cdot 21911^{10}$ | 7.7.30653489.1 | $S_7$ (as 7T7) | $1$ | $14$ | |
14.238...809.30t565.a.a | $14$ | $ 1399^{9} \cdot 21911^{9}$ | 7.7.30653489.1 | $S_7$ (as 7T7) | $1$ | $14$ | |
14.270...449.30t565.a.a | $14$ | $ 1399^{5} \cdot 21911^{5}$ | 7.7.30653489.1 | $S_7$ (as 7T7) | $1$ | $14$ | |
15.270...449.42t412.a.a | $15$ | $ 1399^{5} \cdot 21911^{5}$ | 7.7.30653489.1 | $S_7$ (as 7T7) | $1$ | $15$ | |
15.732...601.42t411.a.a | $15$ | $ 1399^{10} \cdot 21911^{10}$ | 7.7.30653489.1 | $S_7$ (as 7T7) | $1$ | $15$ | |
20.732...601.70.a.a | $20$ | $ 1399^{10} \cdot 21911^{10}$ | 7.7.30653489.1 | $S_7$ (as 7T7) | $1$ | $20$ | |
21.732...601.84.a.a | $21$ | $ 1399^{10} \cdot 21911^{10}$ | 7.7.30653489.1 | $S_7$ (as 7T7) | $1$ | $21$ | |
21.224...889.42t418.a.a | $21$ | $ 1399^{11} \cdot 21911^{11}$ | 7.7.30653489.1 | $S_7$ (as 7T7) | $1$ | $21$ | |
35.536...201.126.a.a | $35$ | $ 1399^{20} \cdot 21911^{20}$ | 7.7.30653489.1 | $S_7$ (as 7T7) | $1$ | $35$ | |
35.198...849.70.a.a | $35$ | $ 1399^{15} \cdot 21911^{15}$ | 7.7.30653489.1 | $S_7$ (as 7T7) | $1$ | $35$ |