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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 135240.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 135240.e1 | 135240bs5 | \([0, -1, 0, -69685856, 223928874156]\) | \(155324313723954725282/13018359375\) | \(3136710578400000000\) | \([2]\) | \(11010048\) | \(2.9916\) | |
| 135240.e2 | 135240bs4 | \([0, -1, 0, -5997616, -5645498564]\) | \(198048499826486404/242568272835\) | \(29222824684303272960\) | \([2]\) | \(5505024\) | \(2.6450\) | |
| 135240.e3 | 135240bs3 | \([0, -1, 0, -4364936, 3483833340]\) | \(76343005935514084/694180580625\) | \(83629722757069440000\) | \([2, 2]\) | \(5505024\) | \(2.6450\) | |
| 135240.e4 | 135240bs6 | \([0, -1, 0, -1277936, 8313136140]\) | \(-957928673903042/123339801817575\) | \(-29718127296585484646400\) | \([2]\) | \(11010048\) | \(2.9916\) | |
| 135240.e5 | 135240bs2 | \([0, -1, 0, -475316, -37050684]\) | \(394315384276816/208332909225\) | \(6274600559977478400\) | \([2, 2]\) | \(2752512\) | \(2.2985\) | |
| 135240.e6 | 135240bs1 | \([0, -1, 0, 112929, -4579560]\) | \(84611246065664/53699121315\) | \(-101082366777414960\) | \([2]\) | \(1376256\) | \(1.9519\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 135240.e have rank \(1\).
Complex multiplication
The elliptic curves in class 135240.e do not have complex multiplication.Modular form 135240.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
