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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 2730a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2730.a3 | 2730a1 | \([1, 1, 0, -22718, 1179252]\) | \(1296772724742600169/140009392373760\) | \(140009392373760\) | \([2]\) | \(12288\) | \(1.4484\) | \(\Gamma_0(N)\)-optimal |
| 2730.a2 | 2730a2 | \([1, 1, 0, -85438, -8366732]\) | \(68973914606086620649/9931302391046400\) | \(9931302391046400\) | \([2, 2]\) | \(24576\) | \(1.7950\) | |
| 2730.a1 | 2730a3 | \([1, 1, 0, -1315758, -581449788]\) | \(251913989442882736925929/6620155222590000\) | \(6620155222590000\) | \([2]\) | \(49152\) | \(2.1415\) | |
| 2730.a4 | 2730a4 | \([1, 1, 0, 141362, -44972252]\) | \(312404265277724598551/1056801141155738160\) | \(-1056801141155738160\) | \([2]\) | \(49152\) | \(2.1415\) |
Rank
sage: E.rank()
The elliptic curves in class 2730a have rank \(1\).
Complex multiplication
The elliptic curves in class 2730a do not have complex multiplication.Modular form 2730.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
