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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 348480w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 348480.w8 | 348480w1 | \([0, 0, 0, 103092, -39120752]\) | \(357911/2160\) | \(-731269251213557760\) | \([2]\) | \(4423680\) | \(2.1097\) | \(\Gamma_0(N)\)-optimal |
| 348480.w6 | 348480w2 | \([0, 0, 0, -1290828, -511380848]\) | \(702595369/72900\) | \(24680337228457574400\) | \([2, 2]\) | \(8847360\) | \(2.4563\) | |
| 348480.w7 | 348480w3 | \([0, 0, 0, -942348, 1152262672]\) | \(-273359449/1536000\) | \(-520013689751863296000\) | \([2]\) | \(13271040\) | \(2.6590\) | |
| 348480.w5 | 348480w4 | \([0, 0, 0, -4775628, 3459897232]\) | \(35578826569/5314410\) | \(1799196583954557173760\) | \([2]\) | \(17694720\) | \(2.8028\) | |
| 348480.w4 | 348480w5 | \([0, 0, 0, -20108748, -34707305072]\) | \(2656166199049/33750\) | \(11426082050211840000\) | \([2]\) | \(17694720\) | \(2.8028\) | |
| 348480.w3 | 348480w6 | \([0, 0, 0, -23245068, 43054613008]\) | \(4102915888729/9000000\) | \(3046955213389824000000\) | \([2, 2]\) | \(26542080\) | \(3.0056\) | |
| 348480.w1 | 348480w7 | \([0, 0, 0, -371725068, 2758550165008]\) | \(16778985534208729/81000\) | \(27422596920508416000\) | \([2]\) | \(53084160\) | \(3.3521\) | |
| 348480.w2 | 348480w8 | \([0, 0, 0, -31608588, 9309482512]\) | \(10316097499609/5859375000\) | \(1983694800384000000000000\) | \([2]\) | \(53084160\) | \(3.3521\) |
Rank
sage: E.rank()
The elliptic curves in class 348480w have rank \(0\).
Complex multiplication
The elliptic curves in class 348480w do not have complex multiplication.Modular form 348480.2.a.w
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
