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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 311170.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 311170.f1 | 311170f3 | \([1, 1, 0, -4436292, -3598333696]\) | \(16232905099479601/4052240\) | \(2410366854289040\) | \([2]\) | \(6967296\) | \(2.3272\) | |
| 311170.f2 | 311170f4 | \([1, 1, 0, -4419472, -3626951244]\) | \(-16048965315233521/256572640900\) | \(-152615390337878428900\) | \([2]\) | \(13934592\) | \(2.6737\) | |
| 311170.f3 | 311170f1 | \([1, 1, 0, -63092, -3361456]\) | \(46694890801/18944000\) | \(11268332993024000\) | \([2]\) | \(2322432\) | \(1.7778\) | \(\Gamma_0(N)\)-optimal |
| 311170.f4 | 311170f2 | \([1, 1, 0, 206028, -24191344]\) | \(1625964918479/1369000000\) | \(-814313126449000000\) | \([2]\) | \(4644864\) | \(2.1244\) |
Rank
sage: E.rank()
The elliptic curves in class 311170.f have rank \(0\).
Complex multiplication
The elliptic curves in class 311170.f do not have complex multiplication.Modular form 311170.2.a.f
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.
