Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 12342.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12342.e1 | 12342b2 | \([1, 1, 0, -134080949, -426032184915]\) | \(150476552140919246594353/42832838728685592576\) | \(75880986611028977069531136\) | \([2]\) | \(3893760\) | \(3.6726\) | |
| 12342.e2 | 12342b1 | \([1, 1, 0, -49826229, 130099519917]\) | \(7722211175253055152433/340131399900069888\) | \(602563522938367710855168\) | \([2]\) | \(1946880\) | \(3.3260\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12342.e have rank \(0\).
Complex multiplication
The elliptic curves in class 12342.e do not have complex multiplication.Modular form 12342.2.a.e
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
