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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 80724j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
80724.h4 | 80724j1 | \([0, -1, 0, 6407, 65170]\) | \(2048000/1323\) | \(-18786677919408\) | \([2]\) | \(172800\) | \(1.2359\) | \(\Gamma_0(N)\)-optimal |
80724.h3 | 80724j2 | \([0, -1, 0, -27228, 562968]\) | \(9826000/5103\) | \(1159406408740608\) | \([2]\) | \(345600\) | \(1.5825\) | |
80724.h2 | 80724j3 | \([0, -1, 0, -108913, 14284126]\) | \(-10061824000/352947\) | \(-5011868187166512\) | \([2]\) | \(518400\) | \(1.7852\) | |
80724.h1 | 80724j4 | \([0, -1, 0, -1757028, 897014520]\) | \(2640279346000/3087\) | \(701369308991232\) | \([2]\) | \(1036800\) | \(2.1318\) |
Rank
sage: E.rank()
The elliptic curves in class 80724j have rank \(0\).
Complex multiplication
The elliptic curves in class 80724j do not have complex multiplication.Modular form 80724.2.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 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.