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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 2574.u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2574.u1 | 2574x4 | \([1, -1, 1, -586580, 173046071]\) | \(30618029936661765625/3678951124992\) | \(2681955370119168\) | \([6]\) | \(27648\) | \(1.9851\) | |
| 2574.u2 | 2574x3 | \([1, -1, 1, -33620, 3176759]\) | \(-5764706497797625/2612665516032\) | \(-1904633161187328\) | \([6]\) | \(13824\) | \(1.6386\) | |
| 2574.u3 | 2574x2 | \([1, -1, 1, -16205, -448315]\) | \(645532578015625/252306960048\) | \(183931773874992\) | \([2]\) | \(9216\) | \(1.4358\) | |
| 2574.u4 | 2574x1 | \([1, -1, 1, 3235, -51739]\) | \(5137417856375/4510142208\) | \(-3287893669632\) | \([2]\) | \(4608\) | \(1.0893\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2574.u have rank \(1\).
Complex multiplication
The elliptic curves in class 2574.u do not have complex multiplication.Modular form 2574.2.a.u
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
