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SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 21780y
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
21780.q3 | 21780y1 | \([0, 0, 0, -1452, 14641]\) | \(16384/5\) | \(103317437520\) | \([2]\) | \(17280\) | \(0.81830\) | \(\Gamma_0(N)\)-optimal |
21780.q4 | 21780y2 | \([0, 0, 0, 3993, 98494]\) | \(21296/25\) | \(-8265395001600\) | \([2]\) | \(34560\) | \(1.1649\) | |
21780.q1 | 21780y3 | \([0, 0, 0, -45012, -3674891]\) | \(488095744/125\) | \(2582935938000\) | \([2]\) | \(51840\) | \(1.3676\) | |
21780.q2 | 21780y4 | \([0, 0, 0, -39567, -4597274]\) | \(-20720464/15625\) | \(-5165871876000000\) | \([2]\) | \(103680\) | \(1.7142\) |
Rank
sage: E.rank()
The elliptic curves in class 21780y have rank \(1\).
Complex multiplication
The elliptic curves in class 21780y do not have complex multiplication.Modular form 21780.2.a.y
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.