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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 68544.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68544.x1 | 68544be4 | \([0, 0, 0, -6529227276, -203067673360144]\) | \(322159999717985454060440834/4250799\) | \(406170169638912\) | \([2]\) | \(17694720\) | \(3.7869\) | |
68544.x2 | 68544be3 | \([0, 0, 0, -409126476, -3155787112336]\) | \(79260902459030376659234/842751810121431609\) | \(80526189471796250930184192\) | \([2]\) | \(17694720\) | \(3.7869\) | |
68544.x3 | 68544be2 | \([0, 0, 0, -408076716, -3172932212560]\) | \(157304700372188331121828/18069292138401\) | \(863273875465458745344\) | \([2, 2]\) | \(8847360\) | \(3.4403\) | |
68544.x4 | 68544be1 | \([0, 0, 0, -25439196, -49844774320]\) | \(-152435594466395827792/1646846627220711\) | \(-19669830717340030058496\) | \([2]\) | \(4423680\) | \(3.0937\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 68544.x have rank \(0\).
Complex multiplication
The elliptic curves in class 68544.x do not have complex multiplication.Modular form 68544.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.