Normalized defining polynomial
\( x^{7} - 2x^{6} + 2x + 2 \)
Invariants
Degree: | $7$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(50808384\) \(\medspace = 2^{6}\cdot 3^{8}\cdot 11^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.61\) | 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: | $2^{6/7}3^{23/12}11^{2/3}\approx 73.58171494182155$ | ||
Ramified primes: | \(2\), \(3\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $4$ | 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: | $a^{6}-2a^{5}+a^{4}-a^{3}-a+1$, $a^{3}-a-1$, $a^{3}-a^{2}-a-1$, $a^{6}-3a^{5}+3a^{4}-3a^{3}+2a^{2}-a+3$ | 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: | \( 38.0037816148 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{2}\cdot 38.0037816148 \cdot 1}{2\cdot\sqrt{50808384}}\cr\approx \mathstrut & 0.841935556192 \end{aligned}\]
Galois group
A non-solvable group of order 2520 |
The 9 conjugacy class representatives for $A_7$ |
Character table for $A_7$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 15 siblings: | deg 15, deg 15 |
Degree 21 sibling: | deg 21 |
Degree 35 sibling: | deg 35 |
Degree 42 siblings: | 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 | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | R | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
\(3\) | 3.3.5.3 | $x^{3} + 9 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ |
3.4.3.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
\(11\) | 11.3.2.1 | $x^{3} + 11$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 6.50808384.7t6.a.a | $6$ | $ 2^{6} \cdot 3^{8} \cdot 11^{2}$ | 7.3.50808384.1 | $A_7$ (as 7T6) | $1$ | $2$ |
10.351...448.70.a.a | $10$ | $ 2^{9} \cdot 3^{18} \cdot 11^{6}$ | 7.3.50808384.1 | $A_7$ (as 7T6) | $0$ | $-2$ | |
10.351...448.70.a.b | $10$ | $ 2^{9} \cdot 3^{18} \cdot 11^{6}$ | 7.3.50808384.1 | $A_7$ (as 7T6) | $0$ | $-2$ | |
14.243...856.15t47.a.a | $14$ | $ 2^{12} \cdot 3^{28} \cdot 11^{10}$ | 7.3.50808384.1 | $A_7$ (as 7T6) | $1$ | $2$ | |
14.247...056.21t33.a.a | $14$ | $ 2^{12} \cdot 3^{24} \cdot 11^{8}$ | 7.3.50808384.1 | $A_7$ (as 7T6) | $1$ | $2$ | |
15.223...504.42t294.a.a | $15$ | $ 2^{12} \cdot 3^{26} \cdot 11^{8}$ | 7.3.50808384.1 | $A_7$ (as 7T6) | $1$ | $-1$ | |
21.118...104.42t299.a.a | $21$ | $ 2^{18} \cdot 3^{44} \cdot 11^{16}$ | 7.3.50808384.1 | $A_7$ (as 7T6) | $1$ | $1$ | |
35.294...824.70.a.a | $35$ | $ 2^{30} \cdot 3^{68} \cdot 11^{24}$ | 7.3.50808384.1 | $A_7$ (as 7T6) | $1$ | $-1$ |