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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 3136.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
3136.o1 | 3136r4 | \([0, 0, 0, -116620, -15327984]\) | \(16581375\) | \(10578455953408\) | \([2]\) | \(7168\) | \(1.5601\) | \(-28\) | |
3136.o2 | 3136r3 | \([0, 0, 0, -6860, -268912]\) | \(-3375\) | \(-10578455953408\) | \([2]\) | \(3584\) | \(1.2135\) | \(-7\) | |
3136.o3 | 3136r2 | \([0, 0, 0, -2380, 44688]\) | \(16581375\) | \(89915392\) | \([2]\) | \(1024\) | \(0.58716\) | \(-28\) | |
3136.o4 | 3136r1 | \([0, 0, 0, -140, 784]\) | \(-3375\) | \(-89915392\) | \([2]\) | \(512\) | \(0.24058\) | \(\Gamma_0(N)\)-optimal | \(-7\) |
Rank
sage: E.rank()
The elliptic curves in class 3136.o have rank \(1\).
Complex multiplication
Each elliptic curve in class 3136.o has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-7}) \).Modular form 3136.2.a.o
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.