Normalized defining polynomial
\( x^{7} - 2x^{6} + 3x^{5} + x^{4} - 5x^{3} + 12x^{2} - 7x + 5 \)
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: | \(-60466176\) \(\medspace = -\,2^{10}\cdot 3^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.93\) | 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: | $2^{11/4}3^{37/18}\approx 64.35491690563994$ | ||
Ramified primes: | \(2\), \(3\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{4}$
Monogenic: | Not computed | |
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: | $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{2}a^{4}+\frac{3}{4}a^{3}-\frac{1}{2}a^{2}+\frac{3}{2}a+\frac{3}{4}$, $\frac{5}{4}a^{6}-\frac{11}{4}a^{5}+\frac{7}{2}a^{4}+\frac{7}{4}a^{3}-8a^{2}+14a-\frac{27}{4}$, $\frac{1}{2}a^{5}-a^{4}+a^{3}+\frac{3}{2}a^{2}-\frac{5}{2}a+\frac{11}{2}$ | 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: | \( 29.8469733914 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{3}\cdot 29.8469733914 \cdot 1}{2\cdot\sqrt{60466176}}\cr\approx \mathstrut & 0.952102381732 \end{aligned}\]
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 | R | R | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
2.4.10.4 | $x^{4} + 4 x^{3} + 8 x^{2} + 10$ | $4$ | $1$ | $10$ | $D_{4}$ | $[2, 3, 7/2]$ | |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.6.10.6 | $x^{6} - 18 x^{4} - 12 x^{3} + 162 x^{2} + 432 x + 360$ | $3$ | $2$ | $10$ | $S_3^2$ | $[3/2, 5/2]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
6.3869835264.14t46.a.a | $6$ | $ 2^{16} \cdot 3^{10}$ | 7.1.60466176.1 | $S_7$ (as 7T7) | $1$ | $0$ | |
* | 6.60466176.7t7.a.a | $6$ | $ 2^{10} \cdot 3^{10}$ | 7.1.60466176.1 | $S_7$ (as 7T7) | $1$ | $0$ |
14.982...456.21t38.a.a | $14$ | $ 2^{32} \cdot 3^{28}$ | 7.1.60466176.1 | $S_7$ (as 7T7) | $1$ | $2$ | |
14.628...184.42t413.a.a | $14$ | $ 2^{38} \cdot 3^{28}$ | 7.1.60466176.1 | $S_7$ (as 7T7) | $1$ | $-2$ | |
14.174...144.30t565.a.a | $14$ | $ 2^{36} \cdot 3^{26}$ | 7.1.60466176.1 | $S_7$ (as 7T7) | $1$ | $0$ | |
14.436...536.30t565.a.a | $14$ | $ 2^{34} \cdot 3^{26}$ | 7.1.60466176.1 | $S_7$ (as 7T7) | $1$ | $0$ | |
15.226...624.42t412.a.a | $15$ | $ 2^{40} \cdot 3^{30}$ | 7.1.60466176.1 | $S_7$ (as 7T7) | $1$ | $-3$ | |
15.144...936.42t411.a.a | $15$ | $ 2^{46} \cdot 3^{30}$ | 7.1.60466176.1 | $S_7$ (as 7T7) | $1$ | $3$ | |
20.560...104.70.a.a | $20$ | $ 2^{62} \cdot 3^{40}$ | 7.1.60466176.1 | $S_7$ (as 7T7) | $1$ | $0$ | |
21.709...616.84.a.a | $21$ | $ 2^{56} \cdot 3^{44}$ | 7.1.60466176.1 | $S_7$ (as 7T7) | $1$ | $-3$ | |
21.709...616.42t418.a.a | $21$ | $ 2^{56} \cdot 3^{44}$ | 7.1.60466176.1 | $S_7$ (as 7T7) | $1$ | $3$ | |
35.257...144.126.a.a | $35$ | $ 2^{100} \cdot 3^{74}$ | 7.1.60466176.1 | $S_7$ (as 7T7) | $1$ | $-1$ | |
35.160...384.70.a.a | $35$ | $ 2^{96} \cdot 3^{74}$ | 7.1.60466176.1 | $S_7$ (as 7T7) | $1$ | $1$ |