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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 2142h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2142.c3 | 2142h1 | \([1, -1, 0, -168, -784]\) | \(721734273/13328\) | \(9716112\) | \([2]\) | \(512\) | \(0.13529\) | \(\Gamma_0(N)\)-optimal |
| 2142.c2 | 2142h2 | \([1, -1, 0, -348, 1340]\) | \(6403769793/2775556\) | \(2023380324\) | \([2, 2]\) | \(1024\) | \(0.48186\) | |
| 2142.c1 | 2142h3 | \([1, -1, 0, -4758, 127466]\) | \(16342588257633/8185058\) | \(5966907282\) | \([2]\) | \(2048\) | \(0.82843\) | |
| 2142.c4 | 2142h4 | \([1, -1, 0, 1182, 8990]\) | \(250404380127/196003234\) | \(-142886357586\) | \([2]\) | \(2048\) | \(0.82843\) |
Rank
sage: E.rank()
The elliptic curves in class 2142h have rank \(1\).
Complex multiplication
The elliptic curves in class 2142h do not have complex multiplication.Modular form 2142.2.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.
