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SageMath
E = EllipticCurve("ev1")
E.isogeny_class()
Elliptic curves in class 92400ev
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 92400.cv4 | 92400ev1 | \([0, -1, 0, -13608, -606288]\) | \(4354703137/1617\) | \(103488000000\) | \([2]\) | \(163840\) | \(1.0809\) | \(\Gamma_0(N)\)-optimal |
| 92400.cv3 | 92400ev2 | \([0, -1, 0, -15608, -414288]\) | \(6570725617/2614689\) | \(167340096000000\) | \([2, 2]\) | \(327680\) | \(1.4275\) | |
| 92400.cv6 | 92400ev3 | \([0, -1, 0, 50392, -3054288]\) | \(221115865823/190238433\) | \(-12175259712000000\) | \([2]\) | \(655360\) | \(1.7741\) | |
| 92400.cv2 | 92400ev4 | \([0, -1, 0, -113608, 14481712]\) | \(2533811507137/58110129\) | \(3719048256000000\) | \([2, 2]\) | \(655360\) | \(1.7741\) | |
| 92400.cv5 | 92400ev5 | \([0, -1, 0, 12392, 44721712]\) | \(3288008303/13504609503\) | \(-864295008192000000\) | \([2]\) | \(1310720\) | \(2.1207\) | |
| 92400.cv1 | 92400ev6 | \([0, -1, 0, -1807608, 936017712]\) | \(10206027697760497/5557167\) | \(355658688000000\) | \([2]\) | \(1310720\) | \(2.1207\) |
Rank
sage: E.rank()
The elliptic curves in class 92400ev have rank \(2\).
Complex multiplication
The elliptic curves in class 92400ev do not have complex multiplication.Modular form 92400.2.a.ev
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
