Show commands:
SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 51984.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51984.i1 | 51984cw3 | \([0, 0, 0, -39993024, 97347427504]\) | \(-50357871050752/19\) | \(-2669086710706176\) | \([]\) | \(1866240\) | \(2.7481\) | |
51984.i2 | 51984cw2 | \([0, 0, 0, -485184, 138387184]\) | \(-89915392/6859\) | \(-963540302564929536\) | \([]\) | \(622080\) | \(2.1988\) | |
51984.i3 | 51984cw1 | \([0, 0, 0, 34656, 109744]\) | \(32768/19\) | \(-2669086710706176\) | \([]\) | \(207360\) | \(1.6495\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 51984.i have rank \(0\).
Complex multiplication
The elliptic curves in class 51984.i do not have complex multiplication.Modular form 51984.2.a.i
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.