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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 138600ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 138600.x6 | 138600ca1 | \([0, 0, 0, 500550, 59084125]\) | \(76102438406144/52315569075\) | \(-9534512463918750000\) | \([2]\) | \(2359296\) | \(2.3304\) | \(\Gamma_0(N)\)-optimal |
| 138600.x5 | 138600ca2 | \([0, 0, 0, -2200575, 493965250]\) | \(404151985581136/197735855625\) | \(576597755002500000000\) | \([2, 2]\) | \(4718592\) | \(2.6770\) | |
| 138600.x2 | 138600ca3 | \([0, 0, 0, -28881075, 59697994750]\) | \(228410605013945764/187597265625\) | \(2188134506250000000000\) | \([2, 2]\) | \(9437184\) | \(3.0235\) | |
| 138600.x4 | 138600ca4 | \([0, 0, 0, -18738075, -30877672250]\) | \(62380825826921284/787768887675\) | \(9188536305841200000000\) | \([2]\) | \(9437184\) | \(3.0235\) | |
| 138600.x1 | 138600ca5 | \([0, 0, 0, -462006075, 3822254869750]\) | \(467508233804095622882/315748125\) | \(7365772260000000000\) | \([2]\) | \(18874368\) | \(3.3701\) | |
| 138600.x3 | 138600ca6 | \([0, 0, 0, -22644075, 86199007750]\) | \(-55043996611705922/105743408203125\) | \(-2466782226562500000000000\) | \([2]\) | \(18874368\) | \(3.3701\) |
Rank
sage: E.rank()
The elliptic curves in class 138600ca have rank \(1\).
Complex multiplication
The elliptic curves in class 138600ca do not have complex multiplication.Modular form 138600.2.a.ca
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
