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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 12495k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12495.d3 | 12495k1 | \([1, 0, 0, -16416, 808191]\) | \(4158523459441/16065\) | \(1890031185\) | \([2]\) | \(18432\) | \(0.99346\) | \(\Gamma_0(N)\)-optimal |
| 12495.d2 | 12495k2 | \([1, 0, 0, -16661, 782760]\) | \(4347507044161/258084225\) | \(30363350987025\) | \([2, 2]\) | \(36864\) | \(1.3400\) | |
| 12495.d1 | 12495k3 | \([1, 0, 0, -49736, -3298695]\) | \(115650783909361/27072079335\) | \(3185003061683415\) | \([2]\) | \(73728\) | \(1.6866\) | |
| 12495.d4 | 12495k4 | \([1, 0, 0, 12494, 3237611]\) | \(1833318007919/39525924375\) | \(-4650185476794375\) | \([2]\) | \(73728\) | \(1.6866\) |
Rank
sage: E.rank()
The elliptic curves in class 12495k have rank \(1\).
Complex multiplication
The elliptic curves in class 12495k do not have complex multiplication.Modular form 12495.2.a.k
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.
