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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 704a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 704.h3 | 704a1 | \([0, 1, 0, -1, 1]\) | \(-4096/11\) | \(-704\) | \([]\) | \(16\) | \(-0.76616\) | \(\Gamma_0(N)\)-optimal |
| 704.h2 | 704a2 | \([0, 1, 0, -41, -199]\) | \(-122023936/161051\) | \(-10307264\) | \([]\) | \(80\) | \(0.038564\) | |
| 704.h1 | 704a3 | \([0, 1, 0, -31281, -2139919]\) | \(-52893159101157376/11\) | \(-704\) | \([]\) | \(400\) | \(0.84328\) |
Rank
sage: E.rank()
The elliptic curves in class 704a have rank \(1\).
Complex multiplication
The elliptic curves in class 704a do not have complex multiplication.Modular form 704.2.a.a
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.
\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
