Show commands:
SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 63504.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
63504.i1 | 63504cd2 | \([0, 0, 0, -43659, -3834054]\) | \(-35937/4\) | \(-1024385060192256\) | \([]\) | \(248832\) | \(1.6176\) | |
63504.i2 | 63504cd1 | \([0, 0, 0, 3381, 7546]\) | \(109503/64\) | \(-2498119335936\) | \([]\) | \(82944\) | \(1.0683\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 63504.i have rank \(1\).
Complex multiplication
The elliptic curves in class 63504.i do not have complex multiplication.Modular form 63504.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.