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SageMath
E = EllipticCurve("cv1")
E.isogeny_class()
Elliptic curves in class 34650cv
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 34650.dk5 | 34650cv1 | \([1, -1, 1, -74891255, -292271176753]\) | \(-4078208988807294650401/880065599546327040\) | \(-10024497219832381440000000\) | \([4]\) | \(8847360\) | \(3.5183\) | \(\Gamma_0(N)\)-optimal |
| 34650.dk4 | 34650cv2 | \([1, -1, 1, -1254539255, -17102255176753]\) | \(19170300594578891358373921/671785075055001600\) | \(7652051870548377600000000\) | \([2, 2]\) | \(17694720\) | \(3.8649\) | |
| 34650.dk3 | 34650cv3 | \([1, -1, 1, -1310987255, -15478923592753]\) | \(21876183941534093095979041/3572502915711058560000\) | \(40693041024271276410000000000\) | \([2, 2]\) | \(35389440\) | \(4.2115\) | |
| 34650.dk1 | 34650cv4 | \([1, -1, 1, -20072459255, -1094578718536753]\) | \(78519570041710065450485106721/96428056919040\) | \(1098375835843440000000\) | \([2]\) | \(35389440\) | \(4.2115\) | |
| 34650.dk6 | 34650cv5 | \([1, -1, 1, 2378544745, -86871367792753]\) | \(130650216943167617311657439/361816948816603087500000\) | \(-4121321182614119543554687500000\) | \([2]\) | \(70778880\) | \(4.5580\) | |
| 34650.dk2 | 34650cv6 | \([1, -1, 1, -5903687255, 159806064607247]\) | \(1997773216431678333214187041/187585177195046990066400\) | \(2136712408987332121225087500000\) | \([2]\) | \(70778880\) | \(4.5580\) |
Rank
sage: E.rank()
The elliptic curves in class 34650cv have rank \(0\).
Complex multiplication
The elliptic curves in class 34650cv do not have complex multiplication.Modular form 34650.2.a.cv
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.
