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SageMath
E = EllipticCurve("dr1")
E.isogeny_class()
Elliptic curves in class 333270.dr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 333270.dr1 | 333270dr3 | \([1, -1, 1, -4012206683, 97819264589271]\) | \(66187969564358252770489/550144842789780\) | \(59370620862358585040112180\) | \([2]\) | \(259522560\) | \(4.1153\) | |
| 333270.dr2 | 333270dr2 | \([1, -1, 1, -256253783, 1458039368031]\) | \(17244079743478944889/1469997007491600\) | \(158639376783060467537619600\) | \([2, 2]\) | \(129761280\) | \(3.7687\) | |
| 333270.dr3 | 333270dr1 | \([1, -1, 1, -54768263, -129908312193]\) | \(168351140229842809/29855318411520\) | \(3221931121169597229093120\) | \([2]\) | \(64880640\) | \(3.4221\) | \(\Gamma_0(N)\)-optimal |
| 333270.dr4 | 333270dr4 | \([1, -1, 1, 275930797, 6723047854887]\) | \(21529289381199961031/193397385415972500\) | \(-20871090578759931277211272500\) | \([2]\) | \(259522560\) | \(4.1153\) |
Rank
sage: E.rank()
The elliptic curves in class 333270.dr have rank \(1\).
Complex multiplication
The elliptic curves in class 333270.dr do not have complex multiplication.Modular form 333270.2.a.dr
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
