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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 212940.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212940.j1 | 212940by4 | \([0, 0, 0, -643383, -79606098]\) | \(1210991472/588245\) | \(14307034224541443840\) | \([2]\) | \(3732480\) | \(2.3691\) | |
| 212940.j2 | 212940by3 | \([0, 0, 0, -529308, -148119543]\) | \(10788913152/8575\) | \(13034834388248400\) | \([2]\) | \(1866240\) | \(2.0225\) | |
| 212940.j3 | 212940by2 | \([0, 0, 0, -339183, 76029382]\) | \(129348709488/6125\) | \(204347785824000\) | \([2]\) | \(1244160\) | \(1.8198\) | |
| 212940.j4 | 212940by1 | \([0, 0, 0, -22308, 1056757]\) | \(588791808/109375\) | \(228066725250000\) | \([2]\) | \(622080\) | \(1.4732\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 212940.j have rank \(0\).
Complex multiplication
The elliptic curves in class 212940.j do not have complex multiplication.Modular form 212940.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
