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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 226512.do
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 226512.do1 | 226512bt4 | \([0, 0, 0, -814935, 262864514]\) | \(181037698000/14480427\) | \(4787457957873347328\) | \([2]\) | \(3317760\) | \(2.3277\) | |
| 226512.do2 | 226512bt3 | \([0, 0, 0, -798600, 274687787]\) | \(2725888000000/19773\) | \(408579138416592\) | \([2]\) | \(1658880\) | \(1.9811\) | |
| 226512.do3 | 226512bt2 | \([0, 0, 0, -161535, -24918982]\) | \(1409938000/4563\) | \(1508599895692032\) | \([2]\) | \(1105920\) | \(1.7784\) | |
| 226512.do4 | 226512bt1 | \([0, 0, 0, -14520, -14641]\) | \(16384000/9477\) | \(195827871075408\) | \([2]\) | \(552960\) | \(1.4318\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 226512.do have rank \(0\).
Complex multiplication
The elliptic curves in class 226512.do do not have complex multiplication.Modular form 226512.2.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 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.
