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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 3234.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3234.d1 | 3234b3 | \([1, 1, 0, -3945, 93381]\) | \(57736239625/255552\) | \(30065437248\) | \([2]\) | \(4320\) | \(0.86288\) | |
| 3234.d2 | 3234b4 | \([1, 1, 0, -1985, 188637]\) | \(-7357983625/127552392\) | \(-15006411366408\) | \([2]\) | \(8640\) | \(1.2095\) | |
| 3234.d3 | 3234b1 | \([1, 1, 0, -270, -1728]\) | \(18609625/1188\) | \(139767012\) | \([2]\) | \(1440\) | \(0.31357\) | \(\Gamma_0(N)\)-optimal |
| 3234.d4 | 3234b2 | \([1, 1, 0, 220, -6726]\) | \(9938375/176418\) | \(-20755401282\) | \([2]\) | \(2880\) | \(0.66015\) |
Rank
sage: E.rank()
The elliptic curves in class 3234.d have rank \(0\).
Complex multiplication
The elliptic curves in class 3234.d do not have complex multiplication.Modular form 3234.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
