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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 76664.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
76664.b1 | 76664b4 | \([0, 0, 0, -409331, 100799470]\) | \(1443468546/7\) | \(36782253799424\) | \([2]\) | \(414720\) | \(1.8034\) | |
76664.b2 | 76664b3 | \([0, 0, 0, -80771, -6990114]\) | \(11090466/2401\) | \(12616313053202432\) | \([2]\) | \(414720\) | \(1.8034\) | |
76664.b3 | 76664b2 | \([0, 0, 0, -26011, 1519590]\) | \(740772/49\) | \(128737888297984\) | \([2, 2]\) | \(207360\) | \(1.4569\) | |
76664.b4 | 76664b1 | \([0, 0, 0, 1369, 101306]\) | \(432/7\) | \(-4597781724928\) | \([2]\) | \(103680\) | \(1.1103\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 76664.b have rank \(1\).
Complex multiplication
The elliptic curves in class 76664.b do not have complex multiplication.Modular form 76664.2.a.b
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.