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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 277350.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 277350.bc1 | 277350bc7 | \([1, 0, 1, -246542001, 1489973337148]\) | \(16778985534208729/81000\) | \(8000475108890625000\) | \([2]\) | \(46448640\) | \(3.2495\) | |
| 277350.bc2 | 277350bc8 | \([1, 0, 1, -20964001, 5026645148]\) | \(10316097499609/5859375000\) | \(578738072113037109375000\) | \([2]\) | \(46448640\) | \(3.2495\) | |
| 277350.bc3 | 277350bc6 | \([1, 0, 1, -15417001, 23254087148]\) | \(4102915888729/9000000\) | \(888941678765625000000\) | \([2, 2]\) | \(23224320\) | \(2.9029\) | |
| 277350.bc4 | 277350bc5 | \([1, 0, 1, -13336876, -18747796852]\) | \(2656166199049/33750\) | \(3333531295371093750\) | \([2]\) | \(15482880\) | \(2.7002\) | |
| 277350.bc5 | 277350bc4 | \([1, 0, 1, -3167376, 1868553148]\) | \(35578826569/5314410\) | \(524911171894313906250\) | \([2]\) | \(15482880\) | \(2.7002\) | |
| 277350.bc6 | 277350bc2 | \([1, 0, 1, -856126, -276286852]\) | \(702595369/72900\) | \(7200427598001562500\) | \([2, 2]\) | \(7741440\) | \(2.3536\) | |
| 277350.bc7 | 277350bc3 | \([1, 0, 1, -625001, 622327148]\) | \(-273359449/1536000\) | \(-151712713176000000000\) | \([2]\) | \(11612160\) | \(2.5563\) | |
| 277350.bc8 | 277350bc1 | \([1, 0, 1, 68374, -21124852]\) | \(357911/2160\) | \(-213346002903750000\) | \([2]\) | \(3870720\) | \(2.0070\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 277350.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 277350.bc do not have complex multiplication.Modular form 277350.2.a.bc
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.
