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SageMath
E = EllipticCurve("dq1")
E.isogeny_class()
Elliptic curves in class 34650dq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 34650.ea4 | 34650dq1 | \([1, -1, 1, -45005, 3194997]\) | \(885012508801/127733760\) | \(1454967360000000\) | \([2]\) | \(147456\) | \(1.6348\) | \(\Gamma_0(N)\)-optimal |
| 34650.ea2 | 34650dq2 | \([1, -1, 1, -693005, 222218997]\) | \(3231355012744321/85377600\) | \(972504225000000\) | \([2, 2]\) | \(294912\) | \(1.9814\) | |
| 34650.ea3 | 34650dq3 | \([1, -1, 1, -666005, 240308997]\) | \(-2868190647517441/527295615000\) | \(-6006226614609375000\) | \([2]\) | \(589824\) | \(2.3280\) | |
| 34650.ea1 | 34650dq4 | \([1, -1, 1, -11088005, 14213888997]\) | \(13235378341603461121/9240\) | \(105249375000\) | \([2]\) | \(589824\) | \(2.3280\) |
Rank
sage: E.rank()
The elliptic curves in class 34650dq have rank \(1\).
Complex multiplication
The elliptic curves in class 34650dq do not have complex multiplication.Modular form 34650.2.a.dq
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.
