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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 129360.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129360.d1 | 129360v4 | \([0, -1, 0, -403776, 98886096]\) | \(60430765429444/2525985\) | \(304311919887360\) | \([2]\) | \(1179648\) | \(1.8597\) | |
| 129360.d2 | 129360v3 | \([0, -1, 0, -123496, -15373280]\) | \(1729010797924/148561875\) | \(17897631776640000\) | \([2]\) | \(1179648\) | \(1.8597\) | |
| 129360.d3 | 129360v2 | \([0, -1, 0, -26476, 1391776]\) | \(68150496976/12006225\) | \(361605213446400\) | \([2, 2]\) | \(589824\) | \(1.5131\) | |
| 129360.d4 | 129360v1 | \([0, -1, 0, 3169, 122970]\) | \(1869154304/4611915\) | \(-8681395005360\) | \([2]\) | \(294912\) | \(1.1665\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 129360.d have rank \(0\).
Complex multiplication
The elliptic curves in class 129360.d do not have complex multiplication.Modular form 129360.2.a.d
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.
