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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 51870a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 51870.d4 | 51870a1 | \([1, 1, 0, 872, 17728]\) | \(73197245859191/172623360000\) | \(-172623360000\) | \([2]\) | \(61440\) | \(0.84018\) | \(\Gamma_0(N)\)-optimal |
| 51870.d3 | 51870a2 | \([1, 1, 0, -7128, 188928]\) | \(40061018056412809/7275103617600\) | \(7275103617600\) | \([2, 2]\) | \(122880\) | \(1.1868\) | |
| 51870.d2 | 51870a3 | \([1, 1, 0, -33728, -2221032]\) | \(4243415895694547209/351514682293320\) | \(351514682293320\) | \([2]\) | \(245760\) | \(1.5333\) | |
| 51870.d1 | 51870a4 | \([1, 1, 0, -108528, 13715688]\) | \(141369383441705190409/6345626621880\) | \(6345626621880\) | \([2]\) | \(245760\) | \(1.5333\) |
Rank
sage: E.rank()
The elliptic curves in class 51870a have rank \(2\).
Complex multiplication
The elliptic curves in class 51870a do not have complex multiplication.Modular form 51870.2.a.a
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.
