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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 129360ef
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129360.bw5 | 129360ef1 | \([0, -1, 0, -260954416, 1901227096000]\) | \(-4078208988807294650401/880065599546327040\) | \(-424095079305321799389020160\) | \([2]\) | \(53084160\) | \(3.8304\) | \(\Gamma_0(N)\)-optimal |
| 129360.bw4 | 129360ef2 | \([0, -1, 0, -4371372336, 111241632102336]\) | \(19170300594578891358373921/671785075055001600\) | \(323726714040917537744486400\) | \([2, 2]\) | \(106168320\) | \(4.1770\) | |
| 129360.bw3 | 129360ef3 | \([0, -1, 0, -4568062256, 100683159844800]\) | \(21876183941534093095979041/3572502915711058560000\) | \(1721554516092888385640202240000\) | \([2, 2]\) | \(212336640\) | \(4.5235\) | |
| 129360.bw1 | 129360ef4 | \([0, -1, 0, -69941369136, 7119520258078656]\) | \(78519570041710065450485106721/96428056919040\) | \(46467745662845488988160\) | \([2]\) | \(212336640\) | \(4.5235\) | |
| 129360.bw6 | 129360ef5 | \([0, -1, 0, 8287907024, 565030485462976]\) | \(130650216943167617311657439/361816948816603087500000\) | \(-174356079457585302082713600000000\) | \([2]\) | \(424673280\) | \(4.8701\) | |
| 129360.bw2 | 129360ef6 | \([0, -1, 0, -20571070256, -1039409537294400]\) | \(1997773216431678333214187041/187585177195046990066400\) | \(90395478064415061337382476185600\) | \([2]\) | \(424673280\) | \(4.8701\) |
Rank
sage: E.rank()
The elliptic curves in class 129360ef have rank \(0\).
Complex multiplication
The elliptic curves in class 129360ef do not have complex multiplication.Modular form 129360.2.a.ef
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.
