Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 121.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
121.b1 | 121b2 | \([0, -1, 1, -887, -10143]\) | \(-32768\) | \(-2357947691\) | \([]\) | \(44\) | \(0.57588\) | \(-11\) | |
121.b2 | 121b1 | \([0, -1, 1, -7, 10]\) | \(-32768\) | \(-1331\) | \([]\) | \(4\) | \(-0.62307\) | \(\Gamma_0(N)\)-optimal | \(-11\) |
Rank
sage: E.rank()
The elliptic curves in class 121.b have rank \(1\).
Complex multiplication
Each elliptic curve in class 121.b has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-11}) \).Modular form 121.2.a.b
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.
\(\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.