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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 3234.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3234.p1 | 3234r3 | \([1, 1, 1, -1972104, -1066786449]\) | \(7209828390823479793/49509306\) | \(5824720341594\) | \([2]\) | \(36864\) | \(2.0492\) | |
3234.p2 | 3234r4 | \([1, 1, 1, -171844, -2403745]\) | \(4770223741048753/2740574865798\) | \(322425892386268902\) | \([2]\) | \(36864\) | \(2.0492\) | |
3234.p3 | 3234r2 | \([1, 1, 1, -123334, -16685089]\) | \(1763535241378513/4612311396\) | \(542633823428004\) | \([2, 2]\) | \(18432\) | \(1.7026\) | |
3234.p4 | 3234r1 | \([1, 1, 1, -4754, -463345]\) | \(-100999381393/723148272\) | \(-85077671052528\) | \([4]\) | \(9216\) | \(1.3560\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3234.p have rank \(0\).
Complex multiplication
The elliptic curves in class 3234.p do not have complex multiplication.Modular form 3234.2.a.p
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.