Normalized defining polynomial
\( x^{7} - x^{6} - 198x^{5} + 907x^{4} + 4302x^{3} - 20582x^{2} - 18973x + 56911 \)
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: | \(9851127637605409\) \(\medspace = 463^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(192.66\) | 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: | $463^{6/7}\approx 192.65651792795916$ | ||
Ramified primes: | \(463\) | 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: | \(463\) | ||
Dirichlet character group: | $\lbrace$$\chi_{463}(1,·)$, $\chi_{463}(34,·)$, $\chi_{463}(308,·)$, $\chi_{463}(118,·)$, $\chi_{463}(230,·)$, $\chi_{463}(412,·)$, $\chi_{463}(286,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{15772661959}a^{6}-\frac{5352968981}{15772661959}a^{5}+\frac{2073694136}{15772661959}a^{4}+\frac{6798115062}{15772661959}a^{3}+\frac{5604448414}{15772661959}a^{2}-\frac{2002912031}{15772661959}a-\frac{546859212}{15772661959}$
Monogenic: | No | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{56952130}{15772661959}a^{6}-\frac{303672859}{15772661959}a^{5}-\frac{9935257718}{15772661959}a^{4}+\frac{94867067375}{15772661959}a^{3}-\frac{170663626555}{15772661959}a^{2}-\frac{416997124704}{15772661959}a+\frac{737321803667}{15772661959}$, $\frac{89137902}{15772661959}a^{6}+\frac{64223714}{15772661959}a^{5}-\frac{17667971495}{15772661959}a^{4}+\frac{50968388309}{15772661959}a^{3}+\frac{494176006396}{15772661959}a^{2}-\frac{1161407008941}{15772661959}a-\frac{3578277169647}{15772661959}$, $\frac{13886134}{15772661959}a^{6}+\frac{263437272}{15772661959}a^{5}-\frac{2247867511}{15772661959}a^{4}-\frac{34398208200}{15772661959}a^{3}+\frac{182983003699}{15772661959}a^{2}+\frac{144794020168}{15772661959}a-\frac{501149326587}{15772661959}$, $\frac{21153976}{15772661959}a^{6}+\frac{13458490}{15772661959}a^{5}-\frac{4138100187}{15772661959}a^{4}+\frac{12091575766}{15772661959}a^{3}+\frac{112632481270}{15772661959}a^{2}-\frac{255619183465}{15772661959}a-\frac{860273211574}{15772661959}$, $\frac{121392770}{15772661959}a^{6}+\frac{610347661}{15772661959}a^{5}-\frac{20291899772}{15772661959}a^{4}-\frac{11790429743}{15772661959}a^{3}+\frac{443817047568}{15772661959}a^{2}+\frac{188073481659}{15772661959}a-\frac{1137662883097}{15772661959}$, $\frac{139695991}{15772661959}a^{6}+\frac{583890183}{15772661959}a^{5}-\frac{24654439439}{15772661959}a^{4}-\frac{862036200}{15772661959}a^{3}+\frac{596587769236}{15772661959}a^{2}+\frac{213811355148}{15772661959}a-\frac{1533031358401}{15772661959}$ | 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: | \( 191858.77331 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 191858.77331 \cdot 1}{2\cdot\sqrt{9851127637605409}}\cr\approx \mathstrut & 0.12371394739 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(463\) | Deg $7$ | $7$ | $1$ | $6$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.463.7t1.a.a | $1$ | $ 463 $ | 7.7.9851127637605409.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.463.7t1.a.b | $1$ | $ 463 $ | 7.7.9851127637605409.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.463.7t1.a.c | $1$ | $ 463 $ | 7.7.9851127637605409.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.463.7t1.a.d | $1$ | $ 463 $ | 7.7.9851127637605409.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.463.7t1.a.e | $1$ | $ 463 $ | 7.7.9851127637605409.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.463.7t1.a.f | $1$ | $ 463 $ | 7.7.9851127637605409.1 | $C_7$ (as 7T1) | $0$ | $1$ |