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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 17689.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
17689.e1 | 17689h4 | \([1, -1, 1, -657810, -205176646]\) | \(16581375\) | \(1898470992842767\) | \([2]\) | \(100800\) | \(1.9926\) | \(-28\) | |
17689.e2 | 17689h3 | \([1, -1, 1, -38695, -3592802]\) | \(-3375\) | \(-1898470992842767\) | \([2]\) | \(50400\) | \(1.6460\) | \(-7\) | |
17689.e3 | 17689h2 | \([1, -1, 1, -13425, 602018]\) | \(16581375\) | \(16136737183\) | \([2]\) | \(14400\) | \(1.0197\) | \(-28\) | |
17689.e4 | 17689h1 | \([1, -1, 1, -790, 10700]\) | \(-3375\) | \(-16136737183\) | \([2]\) | \(7200\) | \(0.67308\) | \(\Gamma_0(N)\)-optimal | \(-7\) |
Rank
sage: E.rank()
The elliptic curves in class 17689.e have rank \(1\).
Complex multiplication
Each elliptic curve in class 17689.e has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-7}) \).Modular form 17689.2.a.e
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.