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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 3648.bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3648.bg1 | 3648l3 | \([0, 1, 0, -7297, 237503]\) | \(1311494070536/171\) | \(5603328\) | \([4]\) | \(3072\) | \(0.71016\) | |
| 3648.bg2 | 3648l2 | \([0, 1, 0, -457, 3575]\) | \(2582630848/29241\) | \(119771136\) | \([2, 2]\) | \(1536\) | \(0.36359\) | |
| 3648.bg3 | 3648l4 | \([0, 1, 0, -97, 9407]\) | \(-3112136/1172889\) | \(-38433226752\) | \([2]\) | \(3072\) | \(0.71016\) | |
| 3648.bg4 | 3648l1 | \([0, 1, 0, -52, -70]\) | \(247673152/124659\) | \(7978176\) | \([2]\) | \(768\) | \(0.017017\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3648.bg have rank \(0\).
Complex multiplication
The elliptic curves in class 3648.bg do not have complex multiplication.Modular form 3648.2.a.bg
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.
