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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 1296.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1296.f1 | 1296k3 | \([0, 0, 0, -17235, -870894]\) | \(-189613868625/128\) | \(-382205952\) | \([]\) | \(1008\) | \(0.96251\) | |
1296.f2 | 1296k4 | \([0, 0, 0, -13635, -1244862]\) | \(-1159088625/2097152\) | \(-507227047723008\) | \([]\) | \(3024\) | \(1.5118\) | |
1296.f3 | 1296k2 | \([0, 0, 0, -675, 7074]\) | \(-140625/8\) | \(-1934917632\) | \([]\) | \(432\) | \(0.53887\) | |
1296.f4 | 1296k1 | \([0, 0, 0, 45, 18]\) | \(3375/2\) | \(-5971968\) | \([]\) | \(144\) | \(-0.010440\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1296.f have rank \(1\).
Complex multiplication
The elliptic curves in class 1296.f do not have complex multiplication.Modular form 1296.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 21 & 7 \\ 3 & 1 & 7 & 21 \\ 21 & 7 & 1 & 3 \\ 7 & 21 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.