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SageMath
E = EllipticCurve("gb1")
E.isogeny_class()
Elliptic curves in class 342720gb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 342720.gb4 | 342720gb1 | \([0, 0, 0, 8052, -467728]\) | \(302111711/669375\) | \(-127919554560000\) | \([2]\) | \(1048576\) | \(1.3910\) | \(\Gamma_0(N)\)-optimal |
| 342720.gb3 | 342720gb2 | \([0, 0, 0, -63948, -5104528]\) | \(151334226289/28676025\) | \(5480073717350400\) | \([2, 2]\) | \(2097152\) | \(1.7376\) | |
| 342720.gb2 | 342720gb3 | \([0, 0, 0, -308748, 61383152]\) | \(17032120495489/1339001685\) | \(255887206872514560\) | \([2]\) | \(4194304\) | \(2.0842\) | |
| 342720.gb1 | 342720gb4 | \([0, 0, 0, -971148, -368347408]\) | \(530044731605089/26309115\) | \(5027750172426240\) | \([2]\) | \(4194304\) | \(2.0842\) |
Rank
sage: E.rank()
The elliptic curves in class 342720gb have rank \(0\).
Complex multiplication
The elliptic curves in class 342720gb do not have complex multiplication.Modular form 342720.2.a.gb
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.
