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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 49725.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
49725.e1 | 49725j4 | \([1, -1, 1, -3511355, 2533439022]\) | \(420339554066191969/244298925\) | \(2782717442578125\) | \([2]\) | \(786432\) | \(2.2882\) | |
49725.e2 | 49725j2 | \([1, -1, 1, -220730, 39145272]\) | \(104413920565969/2472575625\) | \(28164181728515625\) | \([2, 2]\) | \(393216\) | \(1.9416\) | |
49725.e3 | 49725j1 | \([1, -1, 1, -30605, -1161228]\) | \(278317173889/109245825\) | \(1244378225390625\) | \([2]\) | \(196608\) | \(1.5950\) | \(\Gamma_0(N)\)-optimal |
49725.e4 | 49725j3 | \([1, -1, 1, 27895, 122186022]\) | \(210751100351/566398828125\) | \(-6451636651611328125\) | \([2]\) | \(786432\) | \(2.2882\) |
Rank
sage: E.rank()
The elliptic curves in class 49725.e have rank \(1\).
Complex multiplication
The elliptic curves in class 49725.e do not have complex multiplication.Modular form 49725.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.