Normalized defining polynomial
\( x^{7} - x^{6} - 264x^{5} + 151x^{4} + 13288x^{3} - 18556x^{2} - 69425x - 34621 \)
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: | \(55171016309022769\) \(\medspace = 617^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(246.42\) | 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: | $617^{6/7}\approx 246.41832304919535$ | ||
Ramified primes: | \(617\) | 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: | \(617\) | ||
Dirichlet character group: | $\lbrace$$\chi_{617}(1,·)$, $\chi_{617}(451,·)$, $\chi_{617}(420,·)$, $\chi_{617}(408,·)$, $\chi_{617}(555,·)$, $\chi_{617}(142,·)$, $\chi_{617}(491,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{302698346633}a^{6}-\frac{104643875969}{302698346633}a^{5}-\frac{88023045533}{302698346633}a^{4}-\frac{59939005216}{302698346633}a^{3}+\frac{7826727280}{302698346633}a^{2}+\frac{22143033249}{302698346633}a-\frac{83810835942}{302698346633}$
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: | $\frac{206128802}{302698346633}a^{6}-\frac{1244667187}{302698346633}a^{5}-\frac{42716519947}{302698346633}a^{4}+\frac{136926352326}{302698346633}a^{3}+\frac{1956269647416}{302698346633}a^{2}-\frac{4858980210042}{302698346633}a-\frac{8317399809825}{302698346633}$, $\frac{61658538}{302698346633}a^{6}-\frac{965204246}{302698346633}a^{5}-\frac{23692919744}{302698346633}a^{4}+\frac{141162417507}{302698346633}a^{3}+\frac{1365624577097}{302698346633}a^{2}-\frac{5965312404080}{302698346633}a-\frac{4692828754877}{302698346633}$, $\frac{16871430299}{302698346633}a^{6}-\frac{37110528797}{302698346633}a^{5}-\frac{4409342393354}{302698346633}a^{4}+\frac{7837604777948}{302698346633}a^{3}+\frac{214750252196835}{302698346633}a^{2}-\frac{570499790420984}{302698346633}a-\frac{487089553544051}{302698346633}$, $\frac{1378468950}{302698346633}a^{6}+\frac{7439199206}{302698346633}a^{5}-\frac{315037628460}{302698346633}a^{4}-\frac{1809951669266}{302698346633}a^{3}+\frac{6499057649965}{302698346633}a^{2}+\frac{16065731901419}{302698346633}a+\frac{7440512428626}{302698346633}$, $\frac{470666412}{302698346633}a^{6}+\frac{961782331}{302698346633}a^{5}-\frac{125727575080}{302698346633}a^{4}-\frac{387622621661}{302698346633}a^{3}+\frac{5058994421987}{302698346633}a^{2}+\frac{10131099696796}{302698346633}a+\frac{4465886750932}{302698346633}$, $\frac{766474409}{302698346633}a^{6}-\frac{7525917648}{302698346633}a^{5}-\frac{139970167904}{302698346633}a^{4}+\frac{1349726040649}{302698346633}a^{3}-\frac{1060222177424}{302698346633}a^{2}-\frac{6094829603170}{302698346633}a-\frac{3210857385481}{302698346633}$ | 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: | \( 948249.056766 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 948249.056766 \cdot 1}{2\cdot\sqrt{55171016309022769}}\cr\approx \mathstrut & 0.258372865176 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.1.0.1}{1} }^{7}$ | ${\href{/padicField/31.1.0.1}{1} }^{7}$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(617\) | Deg $7$ | $7$ | $1$ | $6$ |