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SageMath
E = EllipticCurve("hp1")
E.isogeny_class()
Elliptic curves in class 141120.hp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 141120.hp1 | 141120qa4 | \([0, 0, 0, -682668, 56751408]\) | \(57960603/31250\) | \(18970093707264000000\) | \([2]\) | \(3317760\) | \(2.3902\) | |
| 141120.hp2 | 141120qa2 | \([0, 0, 0, -400428, -97527248]\) | \(8527173507/200\) | \(166541289062400\) | \([2]\) | \(1105920\) | \(1.8409\) | |
| 141120.hp3 | 141120qa1 | \([0, 0, 0, -24108, -1640912]\) | \(-1860867/320\) | \(-266466062499840\) | \([2]\) | \(552960\) | \(1.4943\) | \(\Gamma_0(N)\)-optimal |
| 141120.hp4 | 141120qa3 | \([0, 0, 0, 164052, 6964272]\) | \(804357/500\) | \(-303521499316224000\) | \([2]\) | \(1658880\) | \(2.0436\) |
Rank
sage: E.rank()
The elliptic curves in class 141120.hp have rank \(0\).
Complex multiplication
The elliptic curves in class 141120.hp do not have complex multiplication.Modular form 141120.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
