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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 51425u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51425.m3 | 51425u1 | \([1, -1, 1, -2080, -15078]\) | \(35937/17\) | \(470570890625\) | \([2]\) | \(46080\) | \(0.93388\) | \(\Gamma_0(N)\)-optimal |
51425.m2 | 51425u2 | \([1, -1, 1, -17205, 862172]\) | \(20346417/289\) | \(7999705140625\) | \([2, 2]\) | \(92160\) | \(1.2805\) | |
51425.m4 | 51425u3 | \([1, -1, 1, -2080, 2314172]\) | \(-35937/83521\) | \(-2311914785640625\) | \([2]\) | \(184320\) | \(1.6270\) | |
51425.m1 | 51425u4 | \([1, -1, 1, -274330, 55372672]\) | \(82483294977/17\) | \(470570890625\) | \([2]\) | \(184320\) | \(1.6270\) |
Rank
sage: E.rank()
The elliptic curves in class 51425u have rank \(1\).
Complex multiplication
The elliptic curves in class 51425u do not have complex multiplication.Modular form 51425.2.a.u
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.