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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 960a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
960.b4 | 960a1 | \([0, -1, 0, 4, 6]\) | \(85184/405\) | \(-25920\) | \([2]\) | \(64\) | \(-0.47077\) | \(\Gamma_0(N)\)-optimal |
960.b3 | 960a2 | \([0, -1, 0, -41, 105]\) | \(1906624/225\) | \(921600\) | \([2, 2]\) | \(128\) | \(-0.12420\) | |
960.b2 | 960a3 | \([0, -1, 0, -161, -639]\) | \(14172488/1875\) | \(61440000\) | \([2]\) | \(256\) | \(0.22238\) | |
960.b1 | 960a4 | \([0, -1, 0, -641, 6465]\) | \(890277128/15\) | \(491520\) | \([2]\) | \(256\) | \(0.22238\) |
Rank
sage: E.rank()
The elliptic curves in class 960a have rank \(1\).
Complex multiplication
The elliptic curves in class 960a do not have complex multiplication.Modular form 960.2.a.a
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.