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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 185130.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 185130.j1 | 185130dz4 | \([1, -1, 0, -2632496895, -39072988127475]\) | \(1562225332123379392365961/393363080510106009600\) | \(508015818665970092192806502400\) | \([2]\) | \(265420800\) | \(4.4099\) | |
| 185130.j2 | 185130dz2 | \([1, -1, 0, -903845520, 10455460495200]\) | \(63229930193881628103961/26218934428500000\) | \(33860913995719070716500000\) | \([2]\) | \(88473600\) | \(3.8606\) | |
| 185130.j3 | 185130dz1 | \([1, -1, 0, -47804400, 215325409536]\) | \(-9354997870579612441/10093752054144000\) | \(-13035757464954929713536000\) | \([2]\) | \(44236800\) | \(3.5140\) | \(\Gamma_0(N)\)-optimal |
| 185130.j4 | 185130dz3 | \([1, -1, 0, 400673025, -3888823689459]\) | \(5508208700580085578359/8246033269590589440\) | \(-10649487838984588005589647360\) | \([2]\) | \(132710400\) | \(4.0633\) |
Rank
sage: E.rank()
The elliptic curves in class 185130.j have rank \(0\).
Complex multiplication
The elliptic curves in class 185130.j do not have complex multiplication.Modular form 185130.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
