Normalized defining polynomial
\( x^{7} - 7462x^{4} - 97006x^{3} - 1559558x^{2} - 9051406x - 62041733 \)
Invariants
| Degree: | $7$ |
| |
| Signature: | $[1, 3]$ |
| |
| Discriminant: |
\(-385348306064778857983\)
\(\medspace = -\,7^{5}\cdot 13^{6}\cdot 41^{6}\)
|
| |
| Root discriminant: | \(872.64\) |
| |
| Galois root discriminant: | $7^{5/6}13^{6/7}41^{6/7}\approx 1100.1277440755782$ | ||
| Ramified primes: |
\(7\), \(13\), \(41\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{7}a^{2}-\frac{3}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{3}+\frac{2}{7}a-\frac{2}{7}$, $\frac{1}{49}a^{4}+\frac{1}{49}a^{3}+\frac{3}{49}a^{2}-\frac{17}{49}a-\frac{5}{49}$, $\frac{1}{49}a^{5}+\frac{2}{49}a^{3}+\frac{1}{49}a^{2}-\frac{2}{49}a-\frac{9}{49}$, $\frac{1}{424098577}a^{6}-\frac{612880}{424098577}a^{5}+\frac{381273}{424098577}a^{4}+\frac{29677444}{424098577}a^{3}+\frac{10484504}{424098577}a^{2}+\frac{102061823}{424098577}a-\frac{61748966}{424098577}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{14}$, which has order $14$ (assuming GRH) |
| |
| Narrow class group: | $C_{14}$, which has order $14$ (assuming GRH) |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{47\cdots 81}{424098577}a^{6}-\frac{11\cdots 70}{424098577}a^{5}-\frac{28\cdots 65}{424098577}a^{4}-\frac{35\cdots 14}{424098577}a^{3}-\frac{39\cdots 99}{424098577}a^{2}-\frac{22\cdots 55}{424098577}a-\frac{12\cdots 81}{424098577}$, $\frac{66\cdots 86}{60585511}a^{6}+\frac{54\cdots 29}{60585511}a^{5}+\frac{25\cdots 64}{1236439}a^{4}+\frac{41\cdots 95}{60585511}a^{3}+\frac{31\cdots 80}{60585511}a^{2}+\frac{39\cdots 34}{60585511}a+\frac{13\cdots 41}{60585511}$, $\frac{24\cdots 53}{8655073}a^{6}-\frac{49\cdots 38}{60585511}a^{5}-\frac{14\cdots 41}{60585511}a^{4}-\frac{10\cdots 02}{60585511}a^{3}-\frac{13\cdots 83}{60585511}a^{2}-\frac{12\cdots 71}{60585511}a-\frac{80\cdots 53}{60585511}$
|
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| Regulator: | \( 9464104.459419131 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{3}\cdot 9464104.459419131 \cdot 14}{2\cdot\sqrt{385348306064778857983}}\cr\approx \mathstrut & 1.67425050568009 \end{aligned}\] (assuming GRH)
Galois group
| A solvable group of order 42 |
| The 7 conjugacy class representatives for $F_7$ |
| Character table for $F_7$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Galois closure: | data not computed |
| Degree 14 sibling: | data not computed |
| Degree 21 sibling: | 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 | ${\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | R | ${\href{/padicField/43.1.0.1}{1} }^{7}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 7.1.6.5a1.1 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ | |
|
\(13\)
| 13.1.7.6a1.1 | $x^{7} + 13$ | $7$ | $1$ | $6$ | $D_{7}$ | $$[\ ]_{7}^{2}$$ |
|
\(41\)
| 41.1.7.6a1.1 | $x^{7} + 41$ | $7$ | $1$ | $6$ | $D_{7}$ | $$[\ ]_{7}^{2}$$ |