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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 160016.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 160016.d1 | 160016c3 | \([0, 0, 0, -48539, 4098122]\) | \(3087708109600113/15562236068\) | \(63742918934528\) | \([4]\) | \(380928\) | \(1.4956\) | |
| 160016.d2 | 160016c2 | \([0, 0, 0, -4699, -14070]\) | \(2801432090673/1600320016\) | \(6554910785536\) | \([2, 2]\) | \(190464\) | \(1.1490\) | |
| 160016.d3 | 160016c1 | \([0, 0, 0, -3419, -76790]\) | \(1079098444593/2560256\) | \(10486808576\) | \([2]\) | \(95232\) | \(0.80246\) | \(\Gamma_0(N)\)-optimal |
| 160016.d4 | 160016c4 | \([0, 0, 0, 18661, -112182]\) | \(175456112547087/102864405412\) | \(-421332604567552\) | \([2]\) | \(380928\) | \(1.4956\) |
Rank
sage: E.rank()
The elliptic curves in class 160016.d have rank \(2\).
Complex multiplication
The elliptic curves in class 160016.d do not have complex multiplication.Modular form 160016.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
