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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 142912.bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 142912.bm1 | 142912p4 | \([0, 0, 0, -3414764, -2428519088]\) | \(16798320881842096017/2132227789307\) | \(558950721600094208\) | \([2]\) | \(2752512\) | \(2.4272\) | |
| 142912.bm2 | 142912p3 | \([0, 0, 0, -1354604, 582039120]\) | \(1048626554636928177/48569076788309\) | \(12732092065594474496\) | \([2]\) | \(2752512\) | \(2.4272\) | |
| 142912.bm3 | 142912p2 | \([0, 0, 0, -231724, -31053360]\) | \(5249244962308257/1448621666569\) | \(379747478161063936\) | \([2, 2]\) | \(1376256\) | \(2.0807\) | |
| 142912.bm4 | 142912p1 | \([0, 0, 0, 37396, -3172528]\) | \(22062729659823/29354283343\) | \(-7695049252667392\) | \([2]\) | \(688128\) | \(1.7341\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 142912.bm have rank \(1\).
Complex multiplication
The elliptic curves in class 142912.bm do not have complex multiplication.Modular form 142912.2.a.bm
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
