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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 12342d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12342.a5 | 12342d1 | \([1, 1, 0, -34366, -1871756]\) | \(2533811507137/625016832\) | \(1107255443914752\) | \([2]\) | \(61440\) | \(1.5972\) | \(\Gamma_0(N)\)-optimal |
| 12342.a4 | 12342d2 | \([1, 1, 0, -189246, 30064500]\) | \(423108074414017/23284318464\) | \(41249590502402304\) | \([2, 2]\) | \(122880\) | \(1.9437\) | |
| 12342.a2 | 12342d3 | \([1, 1, 0, -2986766, 1985530980]\) | \(1663303207415737537/5483698704\) | \(9714706759756944\) | \([2, 2]\) | \(245760\) | \(2.2903\) | |
| 12342.a6 | 12342d4 | \([1, 1, 0, 130194, 121616004]\) | \(137763859017023/3683199928848\) | \(-6525013349149891728\) | \([2]\) | \(245760\) | \(2.2903\) | |
| 12342.a1 | 12342d5 | \([1, 1, 0, -47788226, 127133929344]\) | \(6812873765474836663297/74052\) | \(131187635172\) | \([2]\) | \(491520\) | \(2.6369\) | |
| 12342.a3 | 12342d6 | \([1, 1, 0, -2945626, 2042937736]\) | \(-1595514095015181697/95635786040388\) | \(-169424628753495805668\) | \([2]\) | \(491520\) | \(2.6369\) |
Rank
sage: E.rank()
The elliptic curves in class 12342d have rank \(0\).
Complex multiplication
The elliptic curves in class 12342d do not have complex multiplication.Modular form 12342.2.a.d
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.
