Normalized defining polynomial
\( x^{7} - x^{6} - 414x^{5} + 4381x^{4} - 10434x^{3} - 32702x^{2} + 167651x - 182573 \)
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: | \(817633815294109969\) \(\medspace = 967^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(362.19\) | 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: | $967^{6/7}\approx 362.19043569583914$ | ||
Ramified primes: | \(967\) | 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: | \(967\) | ||
Dirichlet character group: | $\lbrace$$\chi_{967}(1,·)$, $\chi_{967}(226,·)$, $\chi_{967}(97,·)$, $\chi_{967}(648,·)$, $\chi_{967}(706,·)$, $\chi_{967}(792,·)$, $\chi_{967}(431,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{754232009}a^{6}-\frac{24470856}{754232009}a^{5}+\frac{265355916}{754232009}a^{4}-\frac{331307190}{754232009}a^{3}-\frac{139941541}{754232009}a^{2}-\frac{211520441}{754232009}a-\frac{68317810}{754232009}$
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: | $\frac{877523936}{754232009}a^{6}+\frac{1789254072}{754232009}a^{5}-\frac{357941514189}{754232009}a^{4}+\frac{2755877037425}{754232009}a^{3}-\frac{752126606262}{754232009}a^{2}-\frac{31033757504910}{754232009}a+\frac{52495402951812}{754232009}$, $\frac{51096416}{754232009}a^{6}+\frac{328788194}{754232009}a^{5}-\frac{18639059062}{754232009}a^{4}+\frac{85955213465}{754232009}a^{3}+\frac{82662636605}{754232009}a^{2}-\frac{1005316171937}{754232009}a+\frac{1343461389571}{754232009}$, $\frac{767140260}{754232009}a^{6}+\frac{1693560407}{754232009}a^{5}-\frac{312043141545}{754232009}a^{4}+\frac{2360937682686}{754232009}a^{3}-\frac{473448061263}{754232009}a^{2}-\frac{26487725338294}{754232009}a+\frac{44110969359035}{754232009}$, $\frac{130569266}{754232009}a^{6}+\frac{794166878}{754232009}a^{5}-\frac{48098069963}{754232009}a^{4}+\frac{234352034003}{754232009}a^{3}+\frac{185119893776}{754232009}a^{2}-\frac{2668481384909}{754232009}a+\frac{3612647873507}{754232009}$, $\frac{212774240}{754232009}a^{6}+\frac{480309196}{754232009}a^{5}-\frac{86495606395}{754232009}a^{4}+\frac{650865847301}{754232009}a^{3}-\frac{106976254537}{754232009}a^{2}-\frac{7299515284090}{754232009}a+\frac{11968570536899}{754232009}$, $\frac{20381186}{754232009}a^{6}-\frac{100979858}{754232009}a^{5}-\frac{8153506790}{754232009}a^{4}+\frac{121240064652}{754232009}a^{3}-\frac{650374053335}{754232009}a^{2}+\frac{1511084469183}{754232009}a-\frac{1282670306943}{754232009}$ | 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: | \( 3027322.57187 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 3027322.57187 \cdot 1}{2\cdot\sqrt{817633815294109969}}\cr\approx \mathstrut & 0.214268954615 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.1.0.1}{1} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.1.0.1}{1} }^{7}$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(967\) | Deg $7$ | $7$ | $1$ | $6$ |