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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 40656.cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 40656.cf1 | 40656dj4 | \([0, 1, 0, -438544, -111881068]\) | \(1285429208617/614922\) | \(4462067028959232\) | \([2]\) | \(368640\) | \(1.9578\) | |
| 40656.cf2 | 40656dj3 | \([0, 1, 0, -244944, 45802260]\) | \(223980311017/4278582\) | \(31046733850632192\) | \([4]\) | \(368640\) | \(1.9578\) | |
| 40656.cf3 | 40656dj2 | \([0, 1, 0, -31984, -1134124]\) | \(498677257/213444\) | \(1548816654680064\) | \([2, 2]\) | \(184320\) | \(1.6112\) | |
| 40656.cf4 | 40656dj1 | \([0, 1, 0, 6736, -127404]\) | \(4657463/3696\) | \(-26819336011776\) | \([2]\) | \(92160\) | \(1.2646\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 40656.cf have rank \(1\).
Complex multiplication
The elliptic curves in class 40656.cf do not have complex multiplication.Modular form 40656.2.a.cf
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.
