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SageMath
E = EllipticCurve("hp1")
E.isogeny_class()
Elliptic curves in class 92400hp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 92400.hk4 | 92400hp1 | \([0, 1, 0, -1263008, -37752012]\) | \(3481467828171481/2005331497785\) | \(128341215858240000000\) | \([2]\) | \(2949120\) | \(2.5480\) | \(\Gamma_0(N)\)-optimal |
| 92400.hk2 | 92400hp2 | \([0, 1, 0, -14385008, -20954220012]\) | \(5143681768032498601/14238434358225\) | \(911259798926400000000\) | \([2, 2]\) | \(5898240\) | \(2.8946\) | |
| 92400.hk3 | 92400hp3 | \([0, 1, 0, -8715008, -37635360012]\) | \(-1143792273008057401/8897444448004035\) | \(-569436444672258240000000\) | \([2]\) | \(11796480\) | \(3.2412\) | |
| 92400.hk1 | 92400hp4 | \([0, 1, 0, -230007008, -1342717080012]\) | \(21026497979043461623321/161783881875\) | \(10354168440000000000\) | \([2]\) | \(11796480\) | \(3.2412\) |
Rank
sage: E.rank()
The elliptic curves in class 92400hp have rank \(1\).
Complex multiplication
The elliptic curves in class 92400hp do not have complex multiplication.Modular form 92400.2.a.hp
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.
