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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 37200.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 37200.bd1 | 37200bv4 | \([0, -1, 0, -2651408, -1660826688]\) | \(32208729120020809/658986840\) | \(42175157760000000\) | \([2]\) | \(663552\) | \(2.3096\) | |
| 37200.bd2 | 37200bv2 | \([0, -1, 0, -171408, -24026688]\) | \(8702409880009/1120910400\) | \(71738265600000000\) | \([2, 2]\) | \(331776\) | \(1.9630\) | |
| 37200.bd3 | 37200bv1 | \([0, -1, 0, -43408, 3109312]\) | \(141339344329/17141760\) | \(1097072640000000\) | \([2]\) | \(165888\) | \(1.6164\) | \(\Gamma_0(N)\)-optimal |
| 37200.bd4 | 37200bv3 | \([0, -1, 0, 260592, -125978688]\) | \(30579142915511/124675335000\) | \(-7979221440000000000\) | \([2]\) | \(663552\) | \(2.3096\) |
Rank
sage: E.rank()
The elliptic curves in class 37200.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 37200.bd do not have complex multiplication.Modular form 37200.2.a.bd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
