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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 126.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 126.a1 | 126b3 | \([1, -1, 0, -12096, -509036]\) | \(268498407453697/252\) | \(183708\) | \([2]\) | \(128\) | \(0.73841\) | |
| 126.a2 | 126b5 | \([1, -1, 0, -8226, 286474]\) | \(84448510979617/933897762\) | \(680811468498\) | \([2]\) | \(256\) | \(1.0850\) | |
| 126.a3 | 126b4 | \([1, -1, 0, -936, -3668]\) | \(124475734657/63011844\) | \(45935634276\) | \([2, 2]\) | \(128\) | \(0.73841\) | |
| 126.a4 | 126b2 | \([1, -1, 0, -756, -7808]\) | \(65597103937/63504\) | \(46294416\) | \([2, 2]\) | \(64\) | \(0.39183\) | |
| 126.a5 | 126b1 | \([1, -1, 0, -36, -176]\) | \(-7189057/16128\) | \(-11757312\) | \([2]\) | \(32\) | \(0.045260\) | \(\Gamma_0(N)\)-optimal |
| 126.a6 | 126b6 | \([1, -1, 0, 3474, -31010]\) | \(6359387729183/4218578658\) | \(-3075343841682\) | \([2]\) | \(256\) | \(1.0850\) |
Rank
sage: E.rank()
The elliptic curves in class 126.a have rank \(0\).
Complex multiplication
The elliptic curves in class 126.a do not have complex multiplication.Modular form 126.2.a.a
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
