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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 254320x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 254320.x4 | 254320x1 | \([0, 0, 0, 21097, 40512598]\) | \(168055344/114841375\) | \(-709629852958048000\) | \([2]\) | \(2433024\) | \(2.1043\) | \(\Gamma_0(N)\)-optimal |
| 254320.x3 | 254320x2 | \([0, 0, 0, -1649323, 795876522]\) | \(20074621850244/546390625\) | \(13505066405776000000\) | \([2, 2]\) | \(4866048\) | \(2.4509\) | |
| 254320.x1 | 254320x3 | \([0, 0, 0, -26214323, 51660165522]\) | \(40301032281655122/31112125\) | \(1537988738916608000\) | \([2]\) | \(9732096\) | \(2.7975\) | |
| 254320.x2 | 254320x4 | \([0, 0, 0, -3811043, -1725121342]\) | \(123831683830962/45654296875\) | \(2256862701500000000000\) | \([2]\) | \(9732096\) | \(2.7975\) |
Rank
sage: E.rank()
The elliptic curves in class 254320x have rank \(1\).
Complex multiplication
The elliptic curves in class 254320x do not have complex multiplication.Modular form 254320.2.a.x
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.
