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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 25857.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25857.e1 | 25857i4 | \([1, -1, 1, -137936, 19752490]\) | \(82483294977/17\) | \(59818643937\) | \([2]\) | \(73728\) | \(1.4551\) | |
25857.e2 | 25857i2 | \([1, -1, 1, -8651, 308026]\) | \(20346417/289\) | \(1016916946929\) | \([2, 2]\) | \(36864\) | \(1.1086\) | |
25857.e3 | 25857i1 | \([1, -1, 1, -1046, -5300]\) | \(35937/17\) | \(59818643937\) | \([2]\) | \(18432\) | \(0.76200\) | \(\Gamma_0(N)\)-optimal |
25857.e4 | 25857i3 | \([1, -1, 1, -1046, 825166]\) | \(-35937/83521\) | \(-293888997662481\) | \([2]\) | \(73728\) | \(1.4551\) |
Rank
sage: E.rank()
The elliptic curves in class 25857.e have rank \(2\).
Complex multiplication
The elliptic curves in class 25857.e do not have complex multiplication.Modular form 25857.2.a.e
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.