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SageMath
E = EllipticCurve("dm1")
E.isogeny_class()
Elliptic curves in class 37200.dm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 37200.dm1 | 37200ct4 | \([0, 1, 0, -254188408, 1559761215188]\) | \(28379906689597370652529/1357352437500\) | \(86870556000000000000\) | \([2]\) | \(4976640\) | \(3.3032\) | |
| 37200.dm2 | 37200ct3 | \([0, 1, 0, -15860408, 24452239188]\) | \(-6894246873502147249/47925198774000\) | \(-3067212721536000000000\) | \([2]\) | \(2488320\) | \(2.9566\) | |
| 37200.dm3 | 37200ct2 | \([0, 1, 0, -3412408, 1742415188]\) | \(68663623745397169/19216056254400\) | \(1229827600281600000000\) | \([2]\) | \(1658880\) | \(2.7539\) | |
| 37200.dm4 | 37200ct1 | \([0, 1, 0, 555592, 179023188]\) | \(296354077829711/387386634240\) | \(-24792744591360000000\) | \([2]\) | \(829440\) | \(2.4073\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 37200.dm have rank \(1\).
Complex multiplication
The elliptic curves in class 37200.dm do not have complex multiplication.Modular form 37200.2.a.dm
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
