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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 4275g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4275.j4 | 4275g1 | \([1, -1, 0, 5058, 624591]\) | \(1256216039/15582375\) | \(-177492990234375\) | \([2]\) | \(13824\) | \(1.4150\) | \(\Gamma_0(N)\)-optimal |
| 4275.j3 | 4275g2 | \([1, -1, 0, -86067, 9099216]\) | \(6189976379881/456890625\) | \(5204269775390625\) | \([2, 2]\) | \(27648\) | \(1.7616\) | |
| 4275.j2 | 4275g3 | \([1, -1, 0, -278442, -45727659]\) | \(209595169258201/41748046875\) | \(475536346435546875\) | \([2]\) | \(55296\) | \(2.1082\) | |
| 4275.j1 | 4275g4 | \([1, -1, 0, -1351692, 605208591]\) | \(23977812996389881/146611125\) | \(1669992345703125\) | \([2]\) | \(55296\) | \(2.1082\) |
Rank
sage: E.rank()
The elliptic curves in class 4275g have rank \(0\).
Complex multiplication
The elliptic curves in class 4275g do not have complex multiplication.Modular form 4275.2.a.g
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.
