Normalized defining polynomial
\( x^{7} - 3x^{6} + 3x^{5} + 3x^{4} - 9x^{3} + 3x^{2} + x - 3 \)
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: | \(112021056\) \(\medspace = 2^{6}\cdot 3^{6}\cdot 7^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(14.12\) | 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: | $2^{6/7}3^{25/18}7^{4/5}\approx 39.51596134371401$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$
Monogenic: | Not computed | |
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: | $\frac{1}{2}a^{6}-\frac{3}{2}a^{5}+\frac{3}{2}a^{4}+a^{3}-\frac{7}{2}a^{2}+\frac{3}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{5}-a^{4}+a^{3}+a^{2}-\frac{3}{2}a+1$, $\frac{1}{2}a^{5}-\frac{3}{2}a^{4}+a^{3}+2a^{2}-\frac{9}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{6}-a^{5}+\frac{1}{2}a^{4}+2a^{3}-\frac{5}{2}a^{2}-a+\frac{1}{2}$ | 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: | \( 50.0931046712 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{2}\cdot 50.0931046712 \cdot 1}{2\cdot\sqrt{112021056}}\cr\approx \mathstrut & 0.747390969504 \end{aligned}\]
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 | ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.6.6.2 | $x^{6} - 6 x^{5} + 39 x^{4} + 60 x^{3} - 18 x + 9$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.5.4.1 | $x^{5} + 7$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 6.112021056.7t6.a.a | $6$ | $ 2^{6} \cdot 3^{6} \cdot 7^{4}$ | 7.3.112021056.1 | $A_7$ (as 7T6) | $1$ | $2$ |
10.141...528.70.a.a | $10$ | $ 2^{9} \cdot 3^{14} \cdot 7^{8}$ | 7.3.112021056.1 | $A_7$ (as 7T6) | $0$ | $-2$ | |
10.141...528.70.a.b | $10$ | $ 2^{9} \cdot 3^{14} \cdot 7^{8}$ | 7.3.112021056.1 | $A_7$ (as 7T6) | $0$ | $-2$ | |
14.219...744.15t47.a.a | $14$ | $ 2^{12} \cdot 3^{18} \cdot 7^{12}$ | 7.3.112021056.1 | $A_7$ (as 7T6) | $1$ | $2$ | |
14.219...744.21t33.a.a | $14$ | $ 2^{12} \cdot 3^{18} \cdot 7^{12}$ | 7.3.112021056.1 | $A_7$ (as 7T6) | $1$ | $2$ | |
15.197...696.42t294.a.a | $15$ | $ 2^{12} \cdot 3^{20} \cdot 7^{12}$ | 7.3.112021056.1 | $A_7$ (as 7T6) | $1$ | $-1$ | |
21.179...056.42t299.a.a | $21$ | $ 2^{18} \cdot 3^{30} \cdot 7^{16}$ | 7.3.112021056.1 | $A_7$ (as 7T6) | $1$ | $1$ | |
35.354...976.70.a.a | $35$ | $ 2^{30} \cdot 3^{50} \cdot 7^{28}$ | 7.3.112021056.1 | $A_7$ (as 7T6) | $1$ | $-1$ |