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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 784.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
784.f1 | 784h4 | \([0, 0, 0, -29155, -1915998]\) | \(16581375\) | \(165288374272\) | \([2]\) | \(896\) | \(1.2135\) | \(-28\) | |
784.f2 | 784h3 | \([0, 0, 0, -1715, -33614]\) | \(-3375\) | \(-165288374272\) | \([2]\) | \(448\) | \(0.86696\) | \(-7\) | |
784.f3 | 784h2 | \([0, 0, 0, -595, 5586]\) | \(16581375\) | \(1404928\) | \([2]\) | \(128\) | \(0.24058\) | \(-28\) | |
784.f4 | 784h1 | \([0, 0, 0, -35, 98]\) | \(-3375\) | \(-1404928\) | \([2]\) | \(64\) | \(-0.10599\) | \(\Gamma_0(N)\)-optimal | \(-7\) |
Rank
sage: E.rank()
The elliptic curves in class 784.f have rank \(1\).
Complex multiplication
Each elliptic curve in class 784.f has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-7}) \).Modular form 784.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.