Show commands:
SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 215600.bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
215600.bf1 | 215600bd2 | \([0, 1, 0, -26172008, -45403544012]\) | \(90315183328170247/11712800000000\) | \(257119385600000000000000\) | \([2]\) | \(25952256\) | \(3.2205\) | |
215600.bf2 | 215600bd1 | \([0, 1, 0, 2499992, -3714456012]\) | \(78716413996793/317194240000\) | \(-6963047956480000000000\) | \([2]\) | \(12976128\) | \(2.8739\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 215600.bf have rank \(1\).
Complex multiplication
The elliptic curves in class 215600.bf do not have complex multiplication.Modular form 215600.2.a.bf
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.