Normalized defining polynomial
\( x^{7} - x^{5} - 2x^{4} - 2x^{3} + 2x^{2} - x + 4 \)
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)
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Discriminant: | \(30558784\) \(\medspace = 2^{6}\cdot 691^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.73\) | 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: | $2^{3/2}691^{1/2}\approx 74.35052118176442$ | ||
Ramified primes: | \(2\), \(691\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{2}{7}a^{5}+\frac{3}{7}a^{4}-\frac{1}{7}a^{3}+\frac{2}{7}a+\frac{2}{7}$
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: | $\frac{2}{7}a^{6}+\frac{3}{7}a^{5}-\frac{1}{7}a^{4}-\frac{2}{7}a^{3}-a^{2}-\frac{3}{7}a-\frac{3}{7}$, $\frac{2}{7}a^{6}+\frac{3}{7}a^{5}-\frac{1}{7}a^{4}-\frac{2}{7}a^{3}-a^{2}-\frac{10}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{6}-\frac{2}{7}a^{5}+\frac{3}{7}a^{4}-\frac{1}{7}a^{3}+\frac{2}{7}a-\frac{5}{7}$, $\frac{3}{7}a^{6}+\frac{1}{7}a^{5}+\frac{2}{7}a^{4}-\frac{3}{7}a^{3}-a^{2}-\frac{8}{7}a-\frac{15}{7}$ | 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: | \( 18.3119129313 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{2}\cdot 18.3119129313 \cdot 1}{2\cdot\sqrt{30558784}}\cr\approx \mathstrut & 0.523100829115 \end{aligned}\]
Galois group
$\GL(3,2)$ (as 7T5):
A non-solvable group of order 168 |
The 6 conjugacy class representatives for $\GL(3,2)$ |
Character table for $\GL(3,2)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 8 sibling: | 8.0.933839279558656.2 |
Degree 14 siblings: | deg 14, deg 14 |
Degree 21 sibling: | deg 21 |
Degree 24 sibling: | deg 24 |
Degree 28 sibling: | deg 28 |
Degree 42 siblings: | deg 42, deg 42, deg 42 |
Arithmetically equvalently sibling: | 7.3.30558784.1 |
Minimal sibling: | 7.3.30558784.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.6.6.1 | $x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$ | $2$ | $3$ | $6$ | $A_4$ | $[2, 2]^{3}$ | |
\(691\) | $\Q_{691}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
3.30558784.42t37.b.a | $3$ | $ 2^{6} \cdot 691^{2}$ | 7.3.30558784.2 | $\GL(3,2)$ (as 7T5) | $0$ | $-1$ | |
3.30558784.42t37.b.b | $3$ | $ 2^{6} \cdot 691^{2}$ | 7.3.30558784.2 | $\GL(3,2)$ (as 7T5) | $0$ | $-1$ | |
* | 6.30558784.7t5.b.a | $6$ | $ 2^{6} \cdot 691^{2}$ | 7.3.30558784.2 | $\GL(3,2)$ (as 7T5) | $1$ | $2$ |
7.933...656.8t37.b.a | $7$ | $ 2^{12} \cdot 691^{4}$ | 7.3.30558784.2 | $\GL(3,2)$ (as 7T5) | $1$ | $-1$ | |
8.933...656.21t14.b.a | $8$ | $ 2^{12} \cdot 691^{4}$ | 7.3.30558784.2 | $\GL(3,2)$ (as 7T5) | $1$ | $0$ |