Normalized defining polynomial
\( x^{7} - 903x^{5} - 15953x^{4} - 97223x^{3} - 177590x^{2} + 46956x + 94471 \)
Invariants
Degree: | $7$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[7, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(87495801462998035849\) \(\medspace = 7^{12}\cdot 43^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(706.08\) | 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: | $7^{12/7}43^{6/7}\approx 706.0822861068045$ | ||
Ramified primes: | \(7\), \(43\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $7$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(2107=7^{2}\cdot 43\) | ||
Dirichlet character group: | $\lbrace$$\chi_{2107}(64,·)$, $\chi_{2107}(1,·)$, $\chi_{2107}(1282,·)$, $\chi_{2107}(1989,·)$, $\chi_{2107}(428,·)$, $\chi_{2107}(876,·)$, $\chi_{2107}(1982,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{13}a^{5}+\frac{1}{13}a^{4}-\frac{5}{13}a^{3}+\frac{6}{13}a^{2}-\frac{3}{13}a$, $\frac{1}{8319249289}a^{6}-\frac{253763158}{8319249289}a^{5}+\frac{569928042}{8319249289}a^{4}-\frac{1773636079}{8319249289}a^{3}+\frac{1507064764}{8319249289}a^{2}-\frac{134953919}{639942253}a+\frac{264628895}{639942253}$
Monogenic: | No | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{7}$, which has order $7$ (assuming GRH)
Unit group
Rank: | $6$ | 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: | $\frac{51530522}{8319249289}a^{6}+\frac{4580874}{8319249289}a^{5}-\frac{46562300199}{8319249289}a^{4}-\frac{825286570605}{8319249289}a^{3}-\frac{5068111070272}{8319249289}a^{2}-\frac{9780562355835}{8319249289}a-\frac{73972677605}{639942253}$, $\frac{16093190}{8319249289}a^{6}-\frac{35237955}{8319249289}a^{5}-\frac{14634657532}{8319249289}a^{4}-\frac{221770287996}{8319249289}a^{3}-\frac{949343616985}{8319249289}a^{2}+\frac{9736310751}{8319249289}a+\frac{42420825433}{639942253}$, $\frac{4554934}{639942253}a^{6}-\frac{738919735}{8319249289}a^{5}-\frac{44259008581}{8319249289}a^{4}-\frac{392411196046}{8319249289}a^{3}-\frac{847561792359}{8319249289}a^{2}+\frac{179914946625}{8319249289}a+\frac{33689096309}{639942253}$, $\frac{42893184}{8319249289}a^{6}+\frac{118883110}{8319249289}a^{5}-\frac{40227560809}{8319249289}a^{4}-\frac{777389658038}{8319249289}a^{3}-\frac{4850673582411}{8319249289}a^{2}-\frac{6914808422571}{8319249289}a+\frac{614178909996}{639942253}$, $\frac{39820681}{8319249289}a^{6}-\frac{179486785}{8319249289}a^{5}-\frac{35382797614}{8319249289}a^{4}-\frac{475012059480}{8319249289}a^{3}-\frac{1523105190859}{8319249289}a^{2}+\frac{2846501791240}{8319249289}a+\frac{206971931776}{639942253}$, $\frac{38206832}{8319249289}a^{6}-\frac{285114764}{8319249289}a^{5}-\frac{32502463661}{8319249289}a^{4}-\frac{362634710003}{8319249289}a^{3}-\frac{972951342067}{8319249289}a^{2}-\frac{6831200201}{639942253}a+\frac{52890301775}{639942253}$ (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: | \( 8351632.841147239 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 8351632.841147239 \cdot 7}{2\cdot\sqrt{87495801462998035849}}\cr\approx \mathstrut & 0.399996140188314 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 7 |
The 7 conjugacy class representatives for $C_7$ |
Character table for $C_7$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | R | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.1.0.1}{1} }^{7}$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.1.0.1}{1} }^{7}$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | R | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.7.12.1 | $x^{7} + 42 x^{6} + 7$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ |
\(43\) | 43.7.6.7 | $x^{7} + 172$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |