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SageMath
E = EllipticCurve("ff1")
E.isogeny_class()
Elliptic curves in class 344850.ff
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 344850.ff1 | 344850ff4 | \([1, 1, 1, -1401273838, -12892743630469]\) | \(10993009831928446009969/3767761230468750000\) | \(104294044581413269042968750000\) | \([2]\) | \(447897600\) | \(4.2700\) | |
| 344850.ff2 | 344850ff2 | \([1, 1, 1, -1255347838, -17120145798469]\) | \(7903870428425797297009/886464000000\) | \(24537891411000000000000\) | \([2]\) | \(149299200\) | \(3.7207\) | |
| 344850.ff3 | 344850ff1 | \([1, 1, 1, -78259838, -268953990469]\) | \(-1914980734749238129/20440940544000\) | \(-565818329235456000000000\) | \([2]\) | \(74649600\) | \(3.3741\) | \(\Gamma_0(N)\)-optimal |
| 344850.ff4 | 344850ff3 | \([1, 1, 1, 258604162, -1399748358469]\) | \(69096190760262356111/70568821500000000\) | \(-1953390187271273437500000000\) | \([2]\) | \(223948800\) | \(3.9235\) |
Rank
sage: E.rank()
The elliptic curves in class 344850.ff have rank \(1\).
Complex multiplication
The elliptic curves in class 344850.ff do not have complex multiplication.Modular form 344850.2.a.ff
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.
