Show commands:
SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 311696x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
311696.x2 | 311696x1 | \([0, -1, 0, 42552, -510992]\) | \(1174241375/694232\) | \(-5037565280878592\) | \([]\) | \(1244160\) | \(1.7023\) | \(\Gamma_0(N)\)-optimal |
311696.x1 | 311696x2 | \([0, -1, 0, -635048, -204116240]\) | \(-3903264618625/226719878\) | \(-1645150592162029568\) | \([]\) | \(3732480\) | \(2.2516\) |
Rank
sage: E.rank()
The elliptic curves in class 311696x have rank \(1\).
Complex multiplication
The elliptic curves in class 311696x do not have complex multiplication.Modular form 311696.2.a.x
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 Cremona 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 Cremona labels.