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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 260100.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
260100.j1 | 260100j4 | \([0, 0, 0, -975375, -364790250]\) | \(54000\) | \(1900399082508000000\) | \([2]\) | \(4478976\) | \(2.3018\) | \(-12\) | |
260100.j2 | 260100j2 | \([0, 0, 0, -108375, 13510750]\) | \(54000\) | \(2606857452000000\) | \([2]\) | \(1492992\) | \(1.7525\) | \(-12\) | |
260100.j3 | 260100j3 | \([0, 0, 0, 0, -16581375]\) | \(0\) | \(-118774942656750000\) | \([2]\) | \(2239488\) | \(1.9552\) | \(-3\) | |
260100.j4 | 260100j1 | \([0, 0, 0, 0, 614125]\) | \(0\) | \(-162928590750000\) | \([2]\) | \(746496\) | \(1.4059\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
Rank
sage: E.rank()
The elliptic curves in class 260100.j have rank \(0\).
Complex multiplication
Each elliptic curve in class 260100.j has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 260100.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.