Normalized defining polynomial
\( x^{7} - x^{6} - 7x^{5} + 3x^{4} + 13x^{3} + x^{2} - 4x - 1 \)
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: | \(78373945\) \(\medspace = 5\cdot 13\cdot 1205753\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.42\) | 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: | $5^{1/2}13^{1/2}1205753^{1/2}\approx 8852.906020059176$ | ||
Ramified primes: | \(5\), \(13\), \(1205753\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{78373945}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a$, $a^{6}-a^{5}-7a^{4}+3a^{3}+12a^{2}+a-1$, $a^{6}-a^{5}-7a^{4}+3a^{3}+13a^{2}+a-3$, $a^{6}-a^{5}-7a^{4}+3a^{3}+13a^{2}-3$, $4a^{6}-6a^{5}-25a^{4}+24a^{3}+40a^{2}-14a-9$, $2a^{6}-3a^{5}-12a^{4}+11a^{3}+18a^{2}-4a-4$ | 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: | \( 34.6446938558 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 34.6446938558 \cdot 1}{2\cdot\sqrt{78373945}}\cr\approx \mathstrut & 0.250455658486 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | R | ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | R | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.5.0.1 | $x^{5} + 4 x + 3$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(1205753\) | $\Q_{1205753}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.78373945.2t1.a.a | $1$ | $ 5 \cdot 13 \cdot 1205753 $ | \(\Q(\sqrt{78373945}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
6.295...625.14t46.a.a | $6$ | $ 5^{5} \cdot 13^{5} \cdot 1205753^{5}$ | 7.7.78373945.1 | $S_7$ (as 7T7) | $1$ | $6$ | |
* | 6.78373945.7t7.a.a | $6$ | $ 5 \cdot 13 \cdot 1205753 $ | 7.7.78373945.1 | $S_7$ (as 7T7) | $1$ | $6$ |
14.377...625.21t38.a.a | $14$ | $ 5^{4} \cdot 13^{4} \cdot 1205753^{4}$ | 7.7.78373945.1 | $S_7$ (as 7T7) | $1$ | $14$ | |
14.874...625.42t413.a.a | $14$ | $ 5^{10} \cdot 13^{10} \cdot 1205753^{10}$ | 7.7.78373945.1 | $S_7$ (as 7T7) | $1$ | $14$ | |
14.111...625.30t565.a.a | $14$ | $ 5^{9} \cdot 13^{9} \cdot 1205753^{9}$ | 7.7.78373945.1 | $S_7$ (as 7T7) | $1$ | $14$ | |
14.295...625.30t565.a.a | $14$ | $ 5^{5} \cdot 13^{5} \cdot 1205753^{5}$ | 7.7.78373945.1 | $S_7$ (as 7T7) | $1$ | $14$ | |
15.295...625.42t412.a.a | $15$ | $ 5^{5} \cdot 13^{5} \cdot 1205753^{5}$ | 7.7.78373945.1 | $S_7$ (as 7T7) | $1$ | $15$ | |
15.874...625.42t411.a.a | $15$ | $ 5^{10} \cdot 13^{10} \cdot 1205753^{10}$ | 7.7.78373945.1 | $S_7$ (as 7T7) | $1$ | $15$ | |
20.874...625.70.a.a | $20$ | $ 5^{10} \cdot 13^{10} \cdot 1205753^{10}$ | 7.7.78373945.1 | $S_7$ (as 7T7) | $1$ | $20$ | |
21.874...625.84.a.a | $21$ | $ 5^{10} \cdot 13^{10} \cdot 1205753^{10}$ | 7.7.78373945.1 | $S_7$ (as 7T7) | $1$ | $21$ | |
21.685...625.42t418.a.a | $21$ | $ 5^{11} \cdot 13^{11} \cdot 1205753^{11}$ | 7.7.78373945.1 | $S_7$ (as 7T7) | $1$ | $21$ | |
35.764...625.126.a.a | $35$ | $ 5^{20} \cdot 13^{20} \cdot 1205753^{20}$ | 7.7.78373945.1 | $S_7$ (as 7T7) | $1$ | $35$ | |
35.258...625.70.a.a | $35$ | $ 5^{15} \cdot 13^{15} \cdot 1205753^{15}$ | 7.7.78373945.1 | $S_7$ (as 7T7) | $1$ | $35$ |