Normalized defining polynomial
\( x^{7} - x^{6} - 552x^{5} - 5577x^{4} - 5280x^{3} + 59010x^{2} - 42769x - 7019 \)
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: | \(4586881327254923761\) \(\medspace = 1289^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(463.37\) | 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: | $1289^{6/7}\approx 463.3733707325986$ | ||
Ramified primes: | \(1289\) | 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: | \(1289\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1289}(368,·)$, $\chi_{1289}(1,·)$, $\chi_{1289}(979,·)$, $\chi_{1289}(641,·)$, $\chi_{1289}(714,·)$, $\chi_{1289}(1085,·)$, $\chi_{1289}(79,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{4}+\frac{1}{9}a^{2}-\frac{2}{9}$, $\frac{1}{9}a^{5}+\frac{1}{9}a^{3}-\frac{2}{9}a$, $\frac{1}{149599467}a^{6}+\frac{720649}{16622163}a^{5}-\frac{39256}{49866489}a^{4}-\frac{334673}{16622163}a^{3}+\frac{1057564}{49866489}a^{2}+\frac{4380280}{16622163}a-\frac{8925076}{149599467}$
Monogenic: | No | |
Index: | $81$ | |
Inessential primes: | $3$ |
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: | $\frac{202913999}{149599467}a^{6}-\frac{7325521}{16622163}a^{5}-\frac{37241288540}{49866489}a^{4}-\frac{134274764983}{16622163}a^{3}-\frac{682638636004}{49866489}a^{2}+\frac{1008481076723}{16622163}a-\frac{5783122420658}{149599467}$, $\frac{1479277}{49866489}a^{6}+\frac{2205760}{1846907}a^{5}-\frac{445152605}{16622163}a^{4}-\frac{4066457843}{5540721}a^{3}-\frac{5433174498}{1846907}a^{2}+\frac{19320064841}{5540721}a+\frac{26626223435}{49866489}$, $\frac{85789475}{149599467}a^{6}-\frac{2464618}{16622163}a^{5}-\frac{15266360177}{49866489}a^{4}-\frac{58557482071}{16622163}a^{3}-\frac{521420539066}{49866489}a^{2}+\frac{95705576588}{16622163}a+\frac{1545310038286}{149599467}$, $\frac{11696377}{16622163}a^{6}-\frac{4034272}{5540721}a^{5}-\frac{6492472808}{16622163}a^{4}-\frac{21448319912}{5540721}a^{3}-\frac{49152906398}{16622163}a^{2}+\frac{76311562309}{1846907}a-\frac{196366476914}{5540721}$, $\frac{60301237}{49866489}a^{6}-\frac{2166590}{16622163}a^{5}-\frac{11058396167}{16622163}a^{4}-\frac{122331753065}{16622163}a^{3}-\frac{77527465097}{5540721}a^{2}+\frac{915658319227}{16622163}a+\frac{404840190746}{49866489}$, $\frac{15450610}{149599467}a^{6}+\frac{7985237}{5540721}a^{5}-\frac{3405043147}{49866489}a^{4}-\frac{7372891907}{5540721}a^{3}-\frac{235889065610}{49866489}a^{2}+\frac{25058702035}{1846907}a+\frac{292361908559}{149599467}$ (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: | \( 178245209.06 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 178245209.06 \cdot 1}{2\cdot\sqrt{4586881327254923761}}\cr\approx \mathstrut & 5.3264651207 \end{aligned}\] (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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.1.0.1}{1} }^{7}$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.1.0.1}{1} }^{7}$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.1.0.1}{1} }^{7}$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.1.0.1}{1} }^{7}$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(1289\) | Deg $7$ | $7$ | $1$ | $6$ |