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SageMath
E = EllipticCurve("lk1")
E.isogeny_class()
Elliptic curves in class 388080lk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388080.lk4 | 388080lk1 | \([0, 0, 0, -205947, 380682666]\) | \(-2749884201/176619520\) | \(-62046089720962744320\) | \([2]\) | \(9437184\) | \(2.4777\) | \(\Gamma_0(N)\)-optimal |
| 388080.lk3 | 388080lk2 | \([0, 0, 0, -9237627, 10736406954]\) | \(248158561089321/1859334400\) | \(653180514835916390400\) | \([2, 2]\) | \(18874368\) | \(2.8242\) | |
| 388080.lk1 | 388080lk3 | \([0, 0, 0, -147535227, 689749963434]\) | \(1010962818911303721/57392720\) | \(20161949565088235520\) | \([2]\) | \(37748736\) | \(3.1708\) | |
| 388080.lk2 | 388080lk4 | \([0, 0, 0, -15446907, -5510795094]\) | \(1160306142246441/634128110000\) | \(222767956835374325760000\) | \([2]\) | \(37748736\) | \(3.1708\) |
Rank
sage: E.rank()
The elliptic curves in class 388080lk have rank \(1\).
Complex multiplication
The elliptic curves in class 388080lk do not have complex multiplication.Modular form 388080.2.a.lk
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
