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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 330330g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 330330.g3 | 330330g1 | \([1, 1, 0, -4106258, 3200976948]\) | \(4322215988102009089/33729696000\) | \(59754213975456000\) | \([2]\) | \(10321920\) | \(2.3924\) | \(\Gamma_0(N)\)-optimal |
| 330330.g2 | 330330g2 | \([1, 1, 0, -4193378, 3057943332]\) | \(4603199529814599169/381010880250000\) | \(674984016026570250000\) | \([2, 2]\) | \(20643840\) | \(2.7390\) | |
| 330330.g4 | 330330g3 | \([1, 1, 0, 4395202, 13984334808]\) | \(5300342889693913151/50743075195312500\) | \(-89894453036083007812500\) | \([2]\) | \(41287680\) | \(3.0856\) | |
| 330330.g1 | 330330g4 | \([1, 1, 0, -14175878, -17020857168]\) | \(177835127196092079169/32316396108496500\) | \(57250467006364168036500\) | \([2]\) | \(41287680\) | \(3.0856\) |
Rank
sage: E.rank()
The elliptic curves in class 330330g have rank \(1\).
Complex multiplication
The elliptic curves in class 330330g do not have complex multiplication.Modular form 330330.2.a.g
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.
