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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 85680.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85680.e1 | 85680q6 | \([0, 0, 0, -3884403, -2946690542]\) | \(4341492689834008802/66364515\) | \(99081689978880\) | \([2]\) | \(1572864\) | \(2.2345\) | |
| 85680.e2 | 85680q4 | \([0, 0, 0, -252003, 48561298]\) | \(2370900673008004/7350121275\) | \(5486836131302400\) | \([2]\) | \(786432\) | \(1.8879\) | |
| 85680.e3 | 85680q3 | \([0, 0, 0, -243003, -45951302]\) | \(2125842732972004/8287371225\) | \(6186489469977600\) | \([2, 2]\) | \(786432\) | \(1.8879\) | |
| 85680.e4 | 85680q5 | \([0, 0, 0, -129603, -88975262]\) | \(-161254333699202/2197363593915\) | \(-3280646266806343680\) | \([2]\) | \(1572864\) | \(2.2345\) | |
| 85680.e5 | 85680q2 | \([0, 0, 0, -22503, 44998]\) | \(6752700360016/3903125625\) | \(728416916640000\) | \([2, 2]\) | \(393216\) | \(1.5413\) | |
| 85680.e6 | 85680q1 | \([0, 0, 0, 5622, 5623]\) | \(1684801439744/976171875\) | \(-11386068750000\) | \([2]\) | \(196608\) | \(1.1947\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 85680.e have rank \(0\).
Complex multiplication
The elliptic curves in class 85680.e do not have complex multiplication.Modular form 85680.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
