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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 840.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
840.e1 | 840c3 | \([0, -1, 0, -2240, -40068]\) | \(1214399773444/105\) | \(107520\) | \([2]\) | \(256\) | \(0.40600\) | |
840.e2 | 840c2 | \([0, -1, 0, -140, -588]\) | \(1193895376/11025\) | \(2822400\) | \([2, 2]\) | \(128\) | \(0.059426\) | |
840.e3 | 840c4 | \([0, -1, 0, -40, -1508]\) | \(-7086244/972405\) | \(-995742720\) | \([2]\) | \(256\) | \(0.40600\) | |
840.e4 | 840c1 | \([0, -1, 0, -15, 12]\) | \(24918016/13125\) | \(210000\) | \([4]\) | \(64\) | \(-0.28715\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 840.e have rank \(0\).
Complex multiplication
The elliptic curves in class 840.e do not have complex multiplication.Modular form 840.2.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.