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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 303240x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 303240.x5 | 303240x1 | \([0, -1, 0, -5535, -650268]\) | \(-24918016/229635\) | \(-172854094134960\) | \([2]\) | \(884736\) | \(1.4144\) | \(\Gamma_0(N)\)-optimal |
| 303240.x4 | 303240x2 | \([0, -1, 0, -151740, -22639500]\) | \(32082281296/99225\) | \(1195040650809600\) | \([2, 2]\) | \(1769472\) | \(1.7610\) | |
| 303240.x3 | 303240x3 | \([0, -1, 0, -216720, -1300068]\) | \(23366901604/13505625\) | \(650633243218560000\) | \([2, 2]\) | \(3538944\) | \(2.1076\) | |
| 303240.x1 | 303240x4 | \([0, -1, 0, -2426040, -1453629060]\) | \(32779037733124/315\) | \(15175119375360\) | \([2]\) | \(3538944\) | \(2.1076\) | |
| 303240.x2 | 303240x5 | \([0, -1, 0, -2339400, 1373347500]\) | \(14695548366242/57421875\) | \(5532595605600000000\) | \([2]\) | \(7077888\) | \(2.4541\) | |
| 303240.x6 | 303240x6 | \([0, -1, 0, 866280, -11263668]\) | \(746185003198/432360075\) | \(-41657877785807001600\) | \([2]\) | \(7077888\) | \(2.4541\) |
Rank
sage: E.rank()
The elliptic curves in class 303240x have rank \(1\).
Complex multiplication
The elliptic curves in class 303240x do not have complex multiplication.Modular form 303240.2.a.x
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.
