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SageMath
E = EllipticCurve("dr1")
E.isogeny_class()
Elliptic curves in class 135240dr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 135240.p2 | 135240dr1 | \([0, -1, 0, -15378176, 20897495676]\) | \(9733205763526108/1069365234375\) | \(44188410273210000000000\) | \([2]\) | \(11612160\) | \(3.0787\) | \(\Gamma_0(N)\)-optimal |
| 135240.p1 | 135240dr2 | \([0, -1, 0, -58253176, -148698854324]\) | \(264527137402013054/37471584403125\) | \(3096808633694281363200000\) | \([2]\) | \(23224320\) | \(3.4253\) |
Rank
sage: E.rank()
The elliptic curves in class 135240dr have rank \(0\).
Complex multiplication
The elliptic curves in class 135240dr do not have complex multiplication.Modular form 135240.2.a.dr
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
