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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 29575.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 29575.r1 | 29575a4 | \([1, -1, 0, -19731542, -33730697259]\) | \(11264882429818809/24990875\) | \(1884784068248046875\) | \([2]\) | \(774144\) | \(2.7523\) | |
| 29575.r2 | 29575a2 | \([1, -1, 0, -1247167, -514275384]\) | \(2844576388809/129390625\) | \(9758497394775390625\) | \([2, 2]\) | \(387072\) | \(2.4057\) | |
| 29575.r3 | 29575a1 | \([1, -1, 0, -212042, 27094991]\) | \(13980103929/3901625\) | \(294256229134765625\) | \([2]\) | \(193536\) | \(2.0591\) | \(\Gamma_0(N)\)-optimal |
| 29575.r4 | 29575a3 | \([1, -1, 0, 675208, -1957979009]\) | \(451394172711/22216796875\) | \(-1675566173553466796875\) | \([2]\) | \(774144\) | \(2.7523\) |
Rank
sage: E.rank()
The elliptic curves in class 29575.r have rank \(1\).
Complex multiplication
The elliptic curves in class 29575.r do not have complex multiplication.Modular form 29575.2.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 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.
