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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 12240bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12240.o6 | 12240bm1 | \([0, 0, 0, -11523, 584962]\) | \(-56667352321/16711680\) | \(-49900809093120\) | \([2]\) | \(24576\) | \(1.3429\) | \(\Gamma_0(N)\)-optimal |
| 12240.o5 | 12240bm2 | \([0, 0, 0, -195843, 33357058]\) | \(278202094583041/16646400\) | \(49705884057600\) | \([2, 2]\) | \(49152\) | \(1.6895\) | |
| 12240.o4 | 12240bm3 | \([0, 0, 0, -207363, 29212162]\) | \(330240275458561/67652010000\) | \(202007819427840000\) | \([2, 2]\) | \(98304\) | \(2.0360\) | |
| 12240.o2 | 12240bm4 | \([0, 0, 0, -3133443, 2134916098]\) | \(1139466686381936641/4080\) | \(12182814720\) | \([2]\) | \(98304\) | \(2.0360\) | |
| 12240.o3 | 12240bm5 | \([0, 0, 0, -1039683, -382120382]\) | \(41623544884956481/2962701562500\) | \(8846579462400000000\) | \([2, 2]\) | \(196608\) | \(2.3826\) | |
| 12240.o7 | 12240bm6 | \([0, 0, 0, 440637, 175271362]\) | \(3168685387909439/6278181696900\) | \(-18746550096036249600\) | \([2]\) | \(196608\) | \(2.3826\) | |
| 12240.o1 | 12240bm7 | \([0, 0, 0, -16339683, -25422100382]\) | \(161572377633716256481/914742821250\) | \(2731407428367360000\) | \([2]\) | \(393216\) | \(2.7292\) | |
| 12240.o8 | 12240bm8 | \([0, 0, 0, 943197, -1667423198]\) | \(31077313442863199/420227050781250\) | \(-1254791250000000000000\) | \([2]\) | \(393216\) | \(2.7292\) |
Rank
sage: E.rank()
The elliptic curves in class 12240bm have rank \(1\).
Complex multiplication
The elliptic curves in class 12240bm do not have complex multiplication.Modular form 12240.2.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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
