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SageMath
E = EllipticCurve("fj1")
E.isogeny_class()
Elliptic curves in class 121680fj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121680.dd2 | 121680fj1 | \([0, 0, 0, -791427, 269391746]\) | \(3803721481/26000\) | \(374732135571456000\) | \([2]\) | \(2322432\) | \(2.2061\) | \(\Gamma_0(N)\)-optimal |
| 121680.dd3 | 121680fj2 | \([0, 0, 0, -304707, 597148994]\) | \(-217081801/10562500\) | \(-152234930075904000000\) | \([2]\) | \(4644864\) | \(2.5526\) | |
| 121680.dd1 | 121680fj3 | \([0, 0, 0, -5050227, -4192978894]\) | \(988345570681/44994560\) | \(648496444534538895360\) | \([2]\) | \(6967296\) | \(2.7554\) | |
| 121680.dd4 | 121680fj4 | \([0, 0, 0, 2737293, -15956806606]\) | \(157376536199/7722894400\) | \(-111308335050186090086400\) | \([2]\) | \(13934592\) | \(3.1020\) |
Rank
sage: E.rank()
The elliptic curves in class 121680fj have rank \(0\).
Complex multiplication
The elliptic curves in class 121680fj do not have complex multiplication.Modular form 121680.2.a.fj
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
