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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 25410h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25410.h3 | 25410h1 | \([1, 1, 0, -60623, 4963413]\) | \(13908844989649/1980372240\) | \(3508350225866640\) | \([2]\) | \(184320\) | \(1.7086\) | \(\Gamma_0(N)\)-optimal |
25410.h2 | 25410h2 | \([1, 1, 0, -256643, -45178503]\) | \(1055257664218129/115307784900\) | \(204274774725228900\) | \([2, 2]\) | \(368640\) | \(2.0552\) | |
25410.h4 | 25410h3 | \([1, 1, 0, 342307, -224024973]\) | \(2503876820718671/13702874328990\) | \(-24275477749139853390\) | \([2]\) | \(737280\) | \(2.4017\) | |
25410.h1 | 25410h4 | \([1, 1, 0, -3991913, -3071494257]\) | \(3971101377248209009/56495958750\) | \(100086037179108750\) | \([2]\) | \(737280\) | \(2.4017\) |
Rank
sage: E.rank()
The elliptic curves in class 25410h have rank \(1\).
Complex multiplication
The elliptic curves in class 25410h do not have complex multiplication.Modular form 25410.2.a.h
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.