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SageMath
E = EllipticCurve("es1")
E.isogeny_class()
Elliptic curves in class 47040.es
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.es1 | 47040cg7 | \([0, 1, 0, -16725921, 26323353279]\) | \(16778985534208729/81000\) | \(2498119335936000\) | \([2]\) | \(1327104\) | \(2.5768\) | |
| 47040.es2 | 47040cg8 | \([0, 1, 0, -1422241, 88380095]\) | \(10316097499609/5859375000\) | \(180708864000000000000\) | \([2]\) | \(1327104\) | \(2.5768\) | |
| 47040.es3 | 47040cg6 | \([0, 1, 0, -1045921, 410585279]\) | \(4102915888729/9000000\) | \(277568815104000000\) | \([2, 2]\) | \(663552\) | \(2.2303\) | |
| 47040.es4 | 47040cg5 | \([0, 1, 0, -904801, -331564801]\) | \(2656166199049/33750\) | \(1040883056640000\) | \([2]\) | \(442368\) | \(2.0275\) | |
| 47040.es5 | 47040cg4 | \([0, 1, 0, -214881, 32951295]\) | \(35578826569/5314410\) | \(163901609630760960\) | \([2]\) | \(442368\) | \(2.0275\) | |
| 47040.es6 | 47040cg2 | \([0, 1, 0, -58081, -4900225]\) | \(702595369/72900\) | \(2248307402342400\) | \([2, 2]\) | \(221184\) | \(1.6810\) | |
| 47040.es7 | 47040cg3 | \([0, 1, 0, -42401, 10983615]\) | \(-273359449/1536000\) | \(-47371744444416000\) | \([2]\) | \(331776\) | \(1.8837\) | |
| 47040.es8 | 47040cg1 | \([0, 1, 0, 4639, -371841]\) | \(357911/2160\) | \(-66616515624960\) | \([2]\) | \(110592\) | \(1.3344\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 47040.es have rank \(1\).
Complex multiplication
The elliptic curves in class 47040.es do not have complex multiplication.Modular form 47040.2.a.es
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.
