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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 5550.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5550.r1 | 5550m4 | \([1, 0, 1, -8525126, 9580034648]\) | \(4385367890843575421521/24975000000\) | \(390234375000000\) | \([2]\) | \(165888\) | \(2.4115\) | |
| 5550.r2 | 5550m5 | \([1, 0, 1, -7578126, -7995265352]\) | \(3080272010107543650001/15465841417699560\) | \(241653772151555625000\) | \([2]\) | \(331776\) | \(2.7581\) | |
| 5550.r3 | 5550m3 | \([1, 0, 1, -733126, 27074648]\) | \(2788936974993502801/1593609593601600\) | \(24900149900025000000\) | \([2, 2]\) | \(165888\) | \(2.4115\) | |
| 5550.r4 | 5550m2 | \([1, 0, 1, -533126, 149474648]\) | \(1072487167529950801/2554882560000\) | \(39920040000000000\) | \([2, 2]\) | \(82944\) | \(2.0650\) | |
| 5550.r5 | 5550m1 | \([1, 0, 1, -21126, 4066648]\) | \(-66730743078481/419010969600\) | \(-6547046400000000\) | \([2]\) | \(41472\) | \(1.7184\) | \(\Gamma_0(N)\)-optimal |
| 5550.r6 | 5550m6 | \([1, 0, 1, 2911874, 216614648]\) | \(174751791402194852399/102423900876336360\) | \(-1600373451192755625000\) | \([2]\) | \(331776\) | \(2.7581\) |
Rank
sage: E.rank()
The elliptic curves in class 5550.r have rank \(0\).
Complex multiplication
The elliptic curves in class 5550.r do not have complex multiplication.Modular form 5550.2.a.r
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
