Normalized defining polynomial
\( x^{7} - x^{6} - 378x^{5} + 973x^{4} + 13106x^{3} + 9624x^{2} - 64665x - 91125 \)
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: | \(473984589097059769\) \(\medspace = 883^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(335.05\) | 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: | $883^{6/7}\approx 335.04965454876634$ | ||
Ramified primes: | \(883\) | 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: | \(883\) | ||
Dirichlet character group: | $\lbrace$$\chi_{883}(1,·)$, $\chi_{883}(626,·)$, $\chi_{883}(707,·)$, $\chi_{883}(71,·)$, $\chi_{883}(296,·)$, $\chi_{883}(199,·)$, $\chi_{883}(749,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{15}a^{4}-\frac{1}{15}a^{3}-\frac{4}{15}a^{2}+\frac{4}{15}a$, $\frac{1}{135}a^{5}+\frac{4}{135}a^{4}+\frac{11}{135}a^{3}-\frac{61}{135}a^{2}-\frac{1}{9}a$, $\frac{1}{428895}a^{6}+\frac{1318}{428895}a^{5}+\frac{12953}{428895}a^{4}-\frac{50758}{428895}a^{3}+\frac{31609}{142965}a^{2}+\frac{7472}{15885}a+\frac{170}{353}$
Monogenic: | No | |
Index: | $2025$ | |
Inessential primes: | $3$, $5$ |
Class group and class number
Trivial group, which has order $1$ (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: | $\frac{3129724}{428895}a^{6}-\frac{63952277}{428895}a^{5}+\frac{10785424}{85779}a^{4}+\frac{403317499}{85779}a^{3}+\frac{685931647}{142965}a^{2}-\frac{359779949}{15885}a-\frac{12146546}{353}$, $\frac{212345486}{142965}a^{6}-\frac{769504663}{142965}a^{5}-\frac{78247546226}{142965}a^{4}+\frac{411920665006}{142965}a^{3}+\frac{567396771406}{47655}a^{2}-\frac{269183228639}{15885}a-\frac{18209161912}{353}$, $\frac{638234}{428895}a^{6}+\frac{3325379}{428895}a^{5}-\frac{210175868}{428895}a^{4}-\frac{635790737}{428895}a^{3}+\frac{318773981}{142965}a^{2}+\frac{44885794}{5295}a+\frac{1348042}{353}$, $\frac{177799}{428895}a^{6}-\frac{139403}{428895}a^{5}-\frac{59386753}{428895}a^{4}+\frac{267542693}{428895}a^{3}+\frac{440320741}{142965}a^{2}-\frac{58751162}{15885}a-\frac{4752587}{353}$, $\frac{3409246}{428895}a^{6}-\frac{12505217}{428895}a^{5}-\frac{251573882}{85779}a^{4}+\frac{1334881723}{85779}a^{3}+\frac{9176714293}{142965}a^{2}-\frac{1454371889}{15885}a-\frac{98355149}{353}$, $\frac{1732789}{142965}a^{6}-\frac{6199586}{142965}a^{5}-\frac{639550954}{142965}a^{4}+\frac{3344369144}{142965}a^{3}+\frac{4812146174}{47655}a^{2}-\frac{487882862}{3177}a-\frac{166926689}{353}$ (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: | \( 129015047.061 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 129015047.061 \cdot 1}{2\cdot\sqrt{473984589097059769}}\cr\approx \mathstrut & 11.9932870524 \end{aligned}\] (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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.1.0.1}{1} }^{7}$ | ${\href{/padicField/5.1.0.1}{1} }^{7}$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(883\) | Deg $7$ | $7$ | $1$ | $6$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.883.7t1.a.e | $1$ | $ 883 $ | 7.7.473984589097059769.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.883.7t1.a.a | $1$ | $ 883 $ | 7.7.473984589097059769.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.883.7t1.a.f | $1$ | $ 883 $ | 7.7.473984589097059769.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.883.7t1.a.d | $1$ | $ 883 $ | 7.7.473984589097059769.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.883.7t1.a.b | $1$ | $ 883 $ | 7.7.473984589097059769.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.883.7t1.a.c | $1$ | $ 883 $ | 7.7.473984589097059769.1 | $C_7$ (as 7T1) | $0$ | $1$ |