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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 54096cs
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54096.cz4 | 54096cs1 | \([0, 1, 0, -469632, -202226508]\) | \(-23771111713777/22848457968\) | \(-11010450356130742272\) | \([2]\) | \(1105920\) | \(2.3499\) | \(\Gamma_0(N)\)-optimal |
54096.cz3 | 54096cs2 | \([0, 1, 0, -8764352, -9986678220]\) | \(154502321244119857/55101928644\) | \(26553085145243467776\) | \([2, 2]\) | \(2211840\) | \(2.6965\) | |
54096.cz2 | 54096cs3 | \([0, 1, 0, -10026592, -6923474188]\) | \(231331938231569617/90942310746882\) | \(43824217772277434032128\) | \([4]\) | \(4423680\) | \(3.0431\) | |
54096.cz1 | 54096cs4 | \([0, 1, 0, -140217632, -639122076300]\) | \(632678989847546725777/80515134\) | \(38799462399860736\) | \([2]\) | \(4423680\) | \(3.0431\) |
Rank
sage: E.rank()
The elliptic curves in class 54096cs have rank \(0\).
Complex multiplication
The elliptic curves in class 54096cs do not have complex multiplication.Modular form 54096.2.a.cs
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.