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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1386d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1386.c4 | 1386d1 | \([1, -1, 0, 693, -4347]\) | \(50447927375/39517632\) | \(-28808353728\) | \([2]\) | \(768\) | \(0.69498\) | \(\Gamma_0(N)\)-optimal |
| 1386.c3 | 1386d2 | \([1, -1, 0, -3267, -35235]\) | \(5290763640625/2291573592\) | \(1670557148568\) | \([2]\) | \(1536\) | \(1.0416\) | |
| 1386.c2 | 1386d3 | \([1, -1, 0, -7407, 313497]\) | \(-61653281712625/21875235228\) | \(-15947046481212\) | \([6]\) | \(2304\) | \(1.2443\) | |
| 1386.c1 | 1386d4 | \([1, -1, 0, -127197, 17491383]\) | \(312196988566716625/25367712678\) | \(18493062542262\) | \([6]\) | \(4608\) | \(1.5909\) |
Rank
sage: E.rank()
The elliptic curves in class 1386d have rank \(0\).
Complex multiplication
The elliptic curves in class 1386d do not have complex multiplication.Modular form 1386.2.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 Cremona 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 Cremona labels.
