Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 645.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
645.d1 | 645a4 | \([1, 1, 0, -688, 6667]\) | \(36097320816649/80625\) | \(80625\) | \([2]\) | \(176\) | \(0.18664\) | |
645.d2 | 645a3 | \([1, 1, 0, -118, -407]\) | \(184122897769/51282015\) | \(51282015\) | \([2]\) | \(176\) | \(0.18664\) | |
645.d3 | 645a2 | \([1, 1, 0, -43, 88]\) | \(9116230969/416025\) | \(416025\) | \([2, 2]\) | \(88\) | \(-0.15993\) | |
645.d4 | 645a1 | \([1, 1, 0, 2, 7]\) | \(357911/17415\) | \(-17415\) | \([2]\) | \(44\) | \(-0.50650\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 645.d have rank \(0\).
Complex multiplication
The elliptic curves in class 645.d do not have complex multiplication.Modular form 645.2.a.d
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.