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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 121275df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121275.er4 | 121275df1 | \([1, -1, 0, -38817, -21356784]\) | \(-4826809/144375\) | \(-193476308115234375\) | \([2]\) | \(884736\) | \(1.9974\) | \(\Gamma_0(N)\)-optimal |
| 121275.er3 | 121275df2 | \([1, -1, 0, -1416942, -645647409]\) | \(234770924809/1334025\) | \(1787721086984765625\) | \([2, 2]\) | \(1769472\) | \(2.3440\) | |
| 121275.er2 | 121275df3 | \([1, -1, 0, -2243817, 195284466]\) | \(932288503609/527295615\) | \(706626554982178359375\) | \([2]\) | \(3538944\) | \(2.6906\) | |
| 121275.er1 | 121275df4 | \([1, -1, 0, -22640067, -41457716784]\) | \(957681397954009/31185\) | \(41790882552890625\) | \([2]\) | \(3538944\) | \(2.6906\) |
Rank
sage: E.rank()
The elliptic curves in class 121275df have rank \(1\).
Complex multiplication
The elliptic curves in class 121275df do not have complex multiplication.Modular form 121275.2.a.df
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.
