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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 75504s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 75504.bs3 | 75504s1 | \([0, 1, 0, -4759, 124760]\) | \(420616192/117\) | \(3316362192\) | \([2]\) | \(92160\) | \(0.80863\) | \(\Gamma_0(N)\)-optimal |
| 75504.bs2 | 75504s2 | \([0, 1, 0, -5364, 90396]\) | \(37642192/13689\) | \(6208230023424\) | \([2, 2]\) | \(184320\) | \(1.1552\) | |
| 75504.bs4 | 75504s3 | \([0, 1, 0, 16416, 656676]\) | \(269676572/257049\) | \(-466307055092736\) | \([2]\) | \(368640\) | \(1.5018\) | |
| 75504.bs1 | 75504s4 | \([0, 1, 0, -36824, -2665500]\) | \(3044193988/85293\) | \(154728194429952\) | \([2]\) | \(368640\) | \(1.5018\) |
Rank
sage: E.rank()
The elliptic curves in class 75504s have rank \(1\).
Complex multiplication
The elliptic curves in class 75504s do not have complex multiplication.Modular form 75504.2.a.s
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.
