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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 438080do
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
438080.do3 | 438080do1 | \([0, -1, 0, -6573025, -3544299423]\) | \(46694890801/18944000\) | \(12741540863566413824000\) | \([2]\) | \(25214976\) | \(2.9394\) | \(\Gamma_0(N)\)-optimal |
438080.do4 | 438080do2 | \([0, -1, 0, 21464095, -25822594975]\) | \(1625964918479/1369000000\) | \(-920775413968666624000000\) | \([2]\) | \(50429952\) | \(3.2860\) | |
438080.do1 | 438080do3 | \([0, -1, 0, -462176225, -3824211706783]\) | \(16232905099479601/4052240\) | \(2725495225347253207040\) | \([2]\) | \(75644928\) | \(3.4887\) | |
438080.do2 | 438080do4 | \([0, -1, 0, -460423905, -3854650556575]\) | \(-16048965315233521/256572640900\) | \(-172568137061893020870246400\) | \([2]\) | \(151289856\) | \(3.8353\) |
Rank
sage: E.rank()
The elliptic curves in class 438080do have rank \(0\).
Complex multiplication
The elliptic curves in class 438080do do not have complex multiplication.Modular form 438080.2.a.do
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.