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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 176610dd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 176610.q3 | 176610dd1 | \([1, 1, 0, -134577, 18609669]\) | \(453161802241/9208080\) | \(5477180725633680\) | \([2]\) | \(1720320\) | \(1.8109\) | \(\Gamma_0(N)\)-optimal |
| 176610.q2 | 176610dd2 | \([1, 1, 0, -285957, -31133799]\) | \(4347507044161/1817316900\) | \(1080982473767424900\) | \([2, 2]\) | \(3440640\) | \(2.1574\) | |
| 176610.q4 | 176610dd3 | \([1, 1, 0, 950313, -227206221]\) | \(159564039253919/129962883750\) | \(-77304954118911933750\) | \([2]\) | \(6881280\) | \(2.5040\) | |
| 176610.q1 | 176610dd4 | \([1, 1, 0, -3944307, -3015615729]\) | \(11409011759446561/5015376870\) | \(2983263125879985270\) | \([2]\) | \(6881280\) | \(2.5040\) |
Rank
sage: E.rank()
The elliptic curves in class 176610dd have rank \(1\).
Complex multiplication
The elliptic curves in class 176610dd do not have complex multiplication.Modular form 176610.2.a.dd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
