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SageMath
E = EllipticCurve("ct1")
E.isogeny_class()
Elliptic curves in class 137904.ct
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 137904.ct1 | 137904cq4 | \([0, 1, 0, -2390392, -1423295548]\) | \(305612563186948/663\) | \(3276978551808\) | \([2]\) | \(2064384\) | \(2.0753\) | |
| 137904.ct2 | 137904cq3 | \([0, 1, 0, -193392, -8146332]\) | \(161838334948/87947613\) | \(434694481875883008\) | \([4]\) | \(2064384\) | \(2.0753\) | |
| 137904.ct3 | 137904cq2 | \([0, 1, 0, -149452, -22259860]\) | \(298766385232/439569\) | \(543159194962176\) | \([2, 2]\) | \(1032192\) | \(1.7288\) | |
| 137904.ct4 | 137904cq1 | \([0, 1, 0, -6647, -553500]\) | \(-420616192/1456611\) | \(-112492529348784\) | \([2]\) | \(516096\) | \(1.3822\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 137904.ct have rank \(1\).
Complex multiplication
The elliptic curves in class 137904.ct do not have complex multiplication.Modular form 137904.2.a.ct
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
