Normalized defining polynomial
\( x^{7} - 1094x^{4} + 5470x^{3} + 9846x^{2} - 149331x + 647101 \)
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: | \(-9187938467969310847\) \(\medspace = -\,7^{3}\cdot 547^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(511.72\) | 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: | $7^{1/2}547^{6/7}\approx 588.0243353821147$ | ||
Ramified primes: | \(7\), \(547\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{351}a^{5}-\frac{101}{351}a^{4}+\frac{55}{117}a^{3}+\frac{10}{39}a^{2}-\frac{32}{351}a-\frac{2}{27}$, $\frac{1}{728677053}a^{6}-\frac{783631}{728677053}a^{5}+\frac{307257211}{728677053}a^{4}-\frac{63172139}{242892351}a^{3}+\frac{166036963}{728677053}a^{2}+\frac{358772144}{728677053}a-\frac{26734340}{56052081}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{7}$, which has order $7$ (assuming GRH)
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: | $\frac{38\!\cdots\!81}{728677053}a^{6}-\frac{20\!\cdots\!72}{728677053}a^{5}+\frac{10\!\cdots\!00}{728677053}a^{4}-\frac{16\!\cdots\!64}{242892351}a^{3}+\frac{46\!\cdots\!26}{728677053}a^{2}-\frac{20\!\cdots\!38}{728677053}a+\frac{37\!\cdots\!13}{56052081}$, $\frac{91\!\cdots\!59}{728677053}a^{6}+\frac{80\!\cdots\!74}{728677053}a^{5}+\frac{66\!\cdots\!32}{728677053}a^{4}-\frac{16\!\cdots\!81}{242892351}a^{3}+\frac{89\!\cdots\!48}{728677053}a^{2}+\frac{91\!\cdots\!12}{728677053}a-\frac{45\!\cdots\!02}{56052081}$, $\frac{12\!\cdots\!29}{728677053}a^{6}+\frac{31\!\cdots\!57}{56052081}a^{5}-\frac{41\!\cdots\!89}{56052081}a^{4}-\frac{48\!\cdots\!43}{242892351}a^{3}+\frac{19\!\cdots\!98}{728677053}a^{2}+\frac{40\!\cdots\!62}{728677053}a-\frac{39\!\cdots\!28}{56052081}$ (assuming GRH) | 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: | \( 3094572.1658462654 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{3}\cdot 3094572.1658462654 \cdot 7}{2\cdot\sqrt{9187938467969310847}}\cr\approx \mathstrut & 1.77267541276321 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 14 |
The 5 conjugacy class representatives for $D_{7}$ |
Character table for $D_{7}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Galois closure: | deg 14 |
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.2.0.1}{2} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.2.0.1}{2} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.1.0.1}{1} }^{7}$ | ${\href{/padicField/29.1.0.1}{1} }^{7}$ | ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(547\) | Deg $7$ | $7$ | $1$ | $6$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.2094463.7t2.a.b | $2$ | $ 7 \cdot 547^{2}$ | 7.1.9187938467969310847.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.2094463.7t2.a.c | $2$ | $ 7 \cdot 547^{2}$ | 7.1.9187938467969310847.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.2094463.7t2.a.a | $2$ | $ 7 \cdot 547^{2}$ | 7.1.9187938467969310847.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |