Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 26912a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
26912.d3 | 26912a1 | \([0, 0, 0, -841, 0]\) | \(1728\) | \(38068692544\) | \([2, 2]\) | \(11200\) | \(0.71969\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
26912.d4 | 26912a2 | \([0, 0, 0, 3364, 0]\) | \(1728\) | \(-2436396322816\) | \([2]\) | \(22400\) | \(1.0663\) | \(-4\) | |
26912.d1 | 26912a3 | \([0, 0, 0, -9251, -341446]\) | \(287496\) | \(304549540352\) | \([2]\) | \(22400\) | \(1.0663\) | \(-16\) | |
26912.d2 | 26912a4 | \([0, 0, 0, -9251, 341446]\) | \(287496\) | \(304549540352\) | \([2]\) | \(22400\) | \(1.0663\) | \(-16\) |
Rank
sage: E.rank()
The elliptic curves in class 26912a have rank \(1\).
Complex multiplication
Each elliptic curve in class 26912a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 26912.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.