Show commands:
SageMath
E = EllipticCurve("mf1")
E.isogeny_class()
Elliptic curves in class 352800.mf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 352800.mf1 | 352800mf2 | \([0, 0, 0, -3500672700, 79721484136000]\) | \(864335783029582144/59535\) | \(326789504879040000000\) | \([2]\) | \(141557760\) | \(3.8372\) | |
| 352800.mf2 | 352800mf4 | \([0, 0, 0, -245817075, 918566262250]\) | \(2394165105226952/854262178245\) | \(586134026760673901160000000\) | \([2]\) | \(141557760\) | \(3.8372\) | |
| 352800.mf3 | 352800mf1 | \([0, 0, 0, -218805825, 1245483421000]\) | \(13507798771700416/3544416225\) | \(303990830827713225000000\) | \([2, 2]\) | \(70778880\) | \(3.4907\) | \(\Gamma_0(N)\)-optimal |
| 352800.mf4 | 352800mf3 | \([0, 0, 0, -192015075, 1561855387750]\) | \(-1141100604753992/875529151875\) | \(-600725913429909015000000000\) | \([2]\) | \(141557760\) | \(3.8372\) |
Rank
sage: E.rank()
The elliptic curves in class 352800.mf have rank \(1\).
Complex multiplication
The elliptic curves in class 352800.mf do not have complex multiplication.Modular form 352800.2.a.mf
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
