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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 55470.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55470.bd1 | 55470bc7 | \([1, 1, 1, -9861680, 11915842025]\) | \(16778985534208729/81000\) | \(512030406969000\) | \([2]\) | \(1935360\) | \(2.4448\) | |
| 55470.bd2 | 55470bc8 | \([1, 1, 1, -838560, 39877737]\) | \(10316097499609/5859375000\) | \(37039236615234375000\) | \([2]\) | \(1935360\) | \(2.4448\) | |
| 55470.bd3 | 55470bc6 | \([1, 1, 1, -616680, 185786025]\) | \(4102915888729/9000000\) | \(56892267441000000\) | \([2, 2]\) | \(967680\) | \(2.0982\) | |
| 55470.bd4 | 55470bc5 | \([1, 1, 1, -533475, -150195765]\) | \(2656166199049/33750\) | \(213346002903750\) | \([2]\) | \(645120\) | \(1.8955\) | |
| 55470.bd5 | 55470bc4 | \([1, 1, 1, -126695, 14897747]\) | \(35578826569/5314410\) | \(33594315001236090\) | \([2]\) | \(645120\) | \(1.8955\) | |
| 55470.bd6 | 55470bc2 | \([1, 1, 1, -34245, -2223993]\) | \(702595369/72900\) | \(460827366272100\) | \([2, 2]\) | \(322560\) | \(1.5489\) | |
| 55470.bd7 | 55470bc3 | \([1, 1, 1, -25000, 4968617]\) | \(-273359449/1536000\) | \(-9709613643264000\) | \([2]\) | \(483840\) | \(1.7516\) | |
| 55470.bd8 | 55470bc1 | \([1, 1, 1, 2735, -167905]\) | \(357911/2160\) | \(-13654144185840\) | \([2]\) | \(161280\) | \(1.2023\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 55470.bd have rank \(0\).
Complex multiplication
The elliptic curves in class 55470.bd do not have complex multiplication.Modular form 55470.2.a.bd
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
