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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 370d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
370.d3 | 370d1 | \([1, 0, 0, -75, -143]\) | \(46694890801/18944000\) | \(18944000\) | \([6]\) | \(96\) | \(0.094202\) | \(\Gamma_0(N)\)-optimal |
370.d4 | 370d2 | \([1, 0, 0, 245, -975]\) | \(1625964918479/1369000000\) | \(-1369000000\) | \([6]\) | \(192\) | \(0.44078\) | |
370.d1 | 370d3 | \([1, 0, 0, -5275, -147903]\) | \(16232905099479601/4052240\) | \(4052240\) | \([2]\) | \(288\) | \(0.64351\) | |
370.d2 | 370d4 | \([1, 0, 0, -5255, -149075]\) | \(-16048965315233521/256572640900\) | \(-256572640900\) | \([2]\) | \(576\) | \(0.99008\) |
Rank
sage: E.rank()
The elliptic curves in class 370d have rank \(0\).
Complex multiplication
The elliptic curves in class 370d do not have complex multiplication.Modular form 370.2.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.