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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 304.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
304.f1 | 304e3 | \([0, -1, 0, -12309, 529757]\) | \(-50357871050752/19\) | \(-77824\) | \([]\) | \(216\) | \(0.72659\) | |
304.f2 | 304e2 | \([0, -1, 0, -149, 797]\) | \(-89915392/6859\) | \(-28094464\) | \([]\) | \(72\) | \(0.17728\) | |
304.f3 | 304e1 | \([0, -1, 0, 11, -3]\) | \(32768/19\) | \(-77824\) | \([]\) | \(24\) | \(-0.37203\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 304.f have rank \(0\).
Complex multiplication
The elliptic curves in class 304.f do not have complex multiplication.Modular form 304.2.a.f
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.