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SageMath
E = EllipticCurve("ez1")
E.isogeny_class()
Elliptic curves in class 324870ez
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 324870.ez3 | 324870ez1 | \([1, 0, 0, -158271, -13369959]\) | \(3726830856733921/1501644718080\) | \(176666999437393920\) | \([2]\) | \(5308416\) | \(2.0074\) | \(\Gamma_0(N)\)-optimal |
| 324870.ez2 | 324870ez2 | \([1, 0, 0, -1161791, 472534425]\) | \(1474074790091785441/32813650022400\) | \(3860493111485337600\) | \([2, 2]\) | \(10616832\) | \(2.3539\) | |
| 324870.ez1 | 324870ez3 | \([1, 0, 0, -18488191, 30596213465]\) | \(5940441603429810927841/3044264109120\) | \(358154628173858880\) | \([2]\) | \(21233664\) | \(2.7005\) | |
| 324870.ez4 | 324870ez4 | \([1, 0, 0, 108289, 1450750041]\) | \(1193680917131039/7728836230440000\) | \(-909289853675035560000\) | \([2]\) | \(21233664\) | \(2.7005\) |
Rank
sage: E.rank()
The elliptic curves in class 324870ez have rank \(2\).
Complex multiplication
The elliptic curves in class 324870ez do not have complex multiplication.Modular form 324870.2.a.ez
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.
