Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 121680.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121680.e1 | 121680dz7 | \([0, 0, 0, -129796563, -569171178862]\) | \(16778985534208729/81000\) | \(1167434730049536000\) | \([2]\) | \(10616832\) | \(3.0891\) | |
121680.e2 | 121680dz8 | \([0, 0, 0, -11036883, -1920823918]\) | \(10316097499609/5859375000\) | \(84449850264000000000000\) | \([2]\) | \(10616832\) | \(3.0891\) | |
121680.e3 | 121680dz6 | \([0, 0, 0, -8116563, -8883450862]\) | \(4102915888729/9000000\) | \(129714970005504000000\) | \([2, 2]\) | \(5308416\) | \(2.7425\) | |
121680.e4 | 121680dz5 | \([0, 0, 0, -7021443, 7161152258]\) | \(2656166199049/33750\) | \(486431137520640000\) | \([2]\) | \(3538944\) | \(2.5398\) | |
121680.e5 | 121680dz4 | \([0, 0, 0, -1667523, -713879998]\) | \(35578826569/5314410\) | \(76595392638550056960\) | \([2]\) | \(3538944\) | \(2.5398\) | |
121680.e6 | 121680dz2 | \([0, 0, 0, -450723, 105513122]\) | \(702595369/72900\) | \(1050691257044582400\) | \([2, 2]\) | \(1769472\) | \(2.1932\) | |
121680.e7 | 121680dz3 | \([0, 0, 0, -329043, -237746158]\) | \(-273359449/1536000\) | \(-22138021547606016000\) | \([2]\) | \(2654208\) | \(2.3959\) | |
121680.e8 | 121680dz1 | \([0, 0, 0, 35997, 8071778]\) | \(357911/2160\) | \(-31131592801320960\) | \([2]\) | \(884736\) | \(1.8466\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121680.e have rank \(1\).
Complex multiplication
The elliptic curves in class 121680.e do not have complex multiplication.Modular form 121680.2.a.e
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.