Show commands:
SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 33135.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33135.i1 | 33135j2 | \([0, 1, 1, -179665, 61369681]\) | \(-59501707264/116800875\) | \(-1259021782240612875\) | \([]\) | \(582912\) | \(2.1626\) | |
33135.i2 | 33135j1 | \([0, 1, 1, 19145, -1792256]\) | \(71991296/171315\) | \(-1846641274087635\) | \([]\) | \(194304\) | \(1.6133\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 33135.i have rank \(0\).
Complex multiplication
The elliptic curves in class 33135.i do not have complex multiplication.Modular form 33135.2.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.