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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 56350r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 56350.f3 | 56350r1 | \([1, 0, 1, -185001, -29752852]\) | \(380920459249/12622400\) | \(23203324025000000\) | \([2]\) | \(663552\) | \(1.9133\) | \(\Gamma_0(N)\)-optimal |
| 56350.f4 | 56350r2 | \([1, 0, 1, 59999, -102762852]\) | \(12994449551/2489452840\) | \(-4576275580830625000\) | \([2]\) | \(1327104\) | \(2.2599\) | |
| 56350.f1 | 56350r3 | \([1, 0, 1, -2071501, 1137476148]\) | \(534774372149809/5323062500\) | \(9785202813476562500\) | \([2]\) | \(1990656\) | \(2.4626\) | |
| 56350.f2 | 56350r4 | \([1, 0, 1, -540251, 2782038648]\) | \(-9486391169809/1813439640250\) | \(-3333583753683941406250\) | \([2]\) | \(3981312\) | \(2.8092\) |
Rank
sage: E.rank()
The elliptic curves in class 56350r have rank \(1\).
Complex multiplication
The elliptic curves in class 56350r do not have complex multiplication.Modular form 56350.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
