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SageMath
E = EllipticCurve("dm1")
E.isogeny_class()
Elliptic curves in class 28050dm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 28050.dh4 | 28050dm1 | \([1, 0, 0, -166663, 19687817]\) | \(32765849647039657/8229948198912\) | \(128592940608000000\) | \([2]\) | \(344064\) | \(1.9929\) | \(\Gamma_0(N)\)-optimal |
| 28050.dh2 | 28050dm2 | \([1, 0, 0, -2478663, 1501679817]\) | \(107784459654566688937/10704361149504\) | \(167255642961000000\) | \([2, 2]\) | \(688128\) | \(2.3395\) | |
| 28050.dh3 | 28050dm3 | \([1, 0, 0, -2291663, 1737860817]\) | \(-85183593440646799657/34223681512621656\) | \(-534745023634713375000\) | \([2]\) | \(1376256\) | \(2.6861\) | |
| 28050.dh1 | 28050dm4 | \([1, 0, 0, -39657663, 96122234817]\) | \(441453577446719855661097/4354701912\) | \(68042217375000\) | \([2]\) | \(1376256\) | \(2.6861\) |
Rank
sage: E.rank()
The elliptic curves in class 28050dm have rank \(1\).
Complex multiplication
The elliptic curves in class 28050dm do not have complex multiplication.Modular form 28050.2.a.dm
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.
