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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 3330.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3330.d1 | 3330g3 | \([1, -1, 0, -47475, 3993381]\) | \(16232905099479601/4052240\) | \(2954082960\) | \([6]\) | \(6912\) | \(1.1928\) | |
3330.d2 | 3330g4 | \([1, -1, 0, -47295, 4025025]\) | \(-16048965315233521/256572640900\) | \(-187041455216100\) | \([6]\) | \(13824\) | \(1.5394\) | |
3330.d3 | 3330g1 | \([1, -1, 0, -675, 3861]\) | \(46694890801/18944000\) | \(13810176000\) | \([2]\) | \(2304\) | \(0.64351\) | \(\Gamma_0(N)\)-optimal |
3330.d4 | 3330g2 | \([1, -1, 0, 2205, 26325]\) | \(1625964918479/1369000000\) | \(-998001000000\) | \([2]\) | \(4608\) | \(0.99008\) |
Rank
sage: E.rank()
The elliptic curves in class 3330.d have rank \(1\).
Complex multiplication
The elliptic curves in class 3330.d do not have complex multiplication.Modular form 3330.2.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.