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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 7728.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7728.p1 | 7728u2 | \([0, 1, 0, -53544, -4183884]\) | \(4144806984356137/568114785504\) | \(2326998161424384\) | \([2]\) | \(46080\) | \(1.6755\) | |
| 7728.p2 | 7728u1 | \([0, 1, 0, 5336, -344908]\) | \(4101378352343/15049939968\) | \(-61644554108928\) | \([2]\) | \(23040\) | \(1.3289\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 7728.p have rank \(1\).
Complex multiplication
The elliptic curves in class 7728.p do not have complex multiplication.Modular form 7728.2.a.p
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
