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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 14400.o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 14400.o1 | 14400ef7 | \([0, 0, 0, -76802700, -259067446000]\) | \(16778985534208729/81000\) | \(241864704000000000\) | \([2]\) | \(884736\) | \(2.9579\) | |
| 14400.o2 | 14400ef8 | \([0, 0, 0, -6530700, -874294000]\) | \(10316097499609/5859375000\) | \(17496000000000000000000\) | \([2]\) | \(884736\) | \(2.9579\) | |
| 14400.o3 | 14400ef6 | \([0, 0, 0, -4802700, -4043446000]\) | \(4102915888729/9000000\) | \(26873856000000000000\) | \([2, 2]\) | \(442368\) | \(2.6113\) | |
| 14400.o4 | 14400ef5 | \([0, 0, 0, -4154700, 3259514000]\) | \(2656166199049/33750\) | \(100776960000000000\) | \([2]\) | \(294912\) | \(2.4086\) | |
| 14400.o5 | 14400ef4 | \([0, 0, 0, -986700, -324934000]\) | \(35578826569/5314410\) | \(15868743229440000000\) | \([2]\) | \(294912\) | \(2.4086\) | |
| 14400.o6 | 14400ef2 | \([0, 0, 0, -266700, 48026000]\) | \(702595369/72900\) | \(217678233600000000\) | \([2, 2]\) | \(147456\) | \(2.0620\) | |
| 14400.o7 | 14400ef3 | \([0, 0, 0, -194700, -108214000]\) | \(-273359449/1536000\) | \(-4586471424000000000\) | \([2]\) | \(221184\) | \(2.2648\) | |
| 14400.o8 | 14400ef1 | \([0, 0, 0, 21300, 3674000]\) | \(357911/2160\) | \(-6449725440000000\) | \([2]\) | \(73728\) | \(1.7155\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 14400.o have rank \(1\).
Complex multiplication
The elliptic curves in class 14400.o do not have complex multiplication.Modular form 14400.2.a.o
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.
