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SageMath
E = EllipticCurve("gb1")
E.isogeny_class()
Elliptic curves in class 232050gb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 232050.gb4 | 232050gb1 | \([1, 0, 0, -156153088, 2556904865792]\) | \(-26949791983733109138764089/165161952797784563712000\) | \(-2580655512465383808000000000\) | \([2]\) | \(155713536\) | \(3.9435\) | \(\Gamma_0(N)\)-optimal |
| 232050.gb3 | 232050gb2 | \([1, 0, 0, -3920921088, 94308065793792]\) | \(426646307804307769001905914169/998470877001641316000000\) | \(15601107453150645562500000000\) | \([2, 2]\) | \(311427072\) | \(4.2900\) | |
| 232050.gb1 | 232050gb3 | \([1, 0, 0, -62699459088, 6042872447007792]\) | \(1744596788171434949302427839201849/9588363813082031250000\) | \(149818184579406738281250000\) | \([2]\) | \(622854144\) | \(4.6366\) | |
| 232050.gb2 | 232050gb4 | \([1, 0, 0, -5378671088, 17833043043792]\) | \(1101358349464662961278219354169/628567168199833707765102000\) | \(9821362003122401683829718750000\) | \([2]\) | \(622854144\) | \(4.6366\) |
Rank
sage: E.rank()
The elliptic curves in class 232050gb have rank \(1\).
Complex multiplication
The elliptic curves in class 232050gb do not have complex multiplication.Modular form 232050.2.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.
