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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 327015x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 327015.x4 | 327015x1 | \([1, -1, 0, 2250, -374945]\) | \(357911/17415\) | \(-61278922597815\) | \([2]\) | \(675840\) | \(1.3253\) | \(\Gamma_0(N)\)-optimal |
| 327015.x3 | 327015x2 | \([1, -1, 0, -66195, -6274904]\) | \(9116230969/416025\) | \(1463885373170025\) | \([2, 2]\) | \(1351680\) | \(1.6719\) | |
| 327015.x2 | 327015x3 | \([1, -1, 0, -180270, 21262801]\) | \(184122897769/51282015\) | \(180448270332758415\) | \([2]\) | \(2703360\) | \(2.0184\) | |
| 327015.x1 | 327015x4 | \([1, -1, 0, -1047240, -412231325]\) | \(36097320816649/80625\) | \(283698715730625\) | \([2]\) | \(2703360\) | \(2.0184\) |
Rank
sage: E.rank()
The elliptic curves in class 327015x have rank \(1\).
Complex multiplication
The elliptic curves in class 327015x do not have complex multiplication.Modular form 327015.2.a.x
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.
