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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 56784f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 56784.j3 | 56784f1 | \([0, -1, 0, -6782364, 6800865408]\) | \(27923315228972368/120393\) | \(148765188079872\) | \([2]\) | \(1354752\) | \(2.3490\) | \(\Gamma_0(N)\)-optimal |
| 56784.j2 | 56784f2 | \([0, -1, 0, -6785744, 6793751184]\) | \(6991270724335972/14494474449\) | \(71641149154000118784\) | \([2, 2]\) | \(2709504\) | \(2.6956\) | |
| 56784.j4 | 56784f3 | \([0, -1, 0, -4467064, 11505308944]\) | \(-997241325462146/5206220835543\) | \(-51465079982052295243776\) | \([2]\) | \(5419008\) | \(3.0422\) | |
| 56784.j1 | 56784f4 | \([0, -1, 0, -9158504, 1626829008]\) | \(8594236719188066/4858291807551\) | \(48025695480449914386432\) | \([2]\) | \(5419008\) | \(3.0422\) |
Rank
sage: E.rank()
The elliptic curves in class 56784f have rank \(0\).
Complex multiplication
The elliptic curves in class 56784f do not have complex multiplication.Modular form 56784.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
