Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 900e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
900.b3 | 900e1 | \([0, 0, 0, -300, -1375]\) | \(16384/5\) | \(911250000\) | \([2]\) | \(288\) | \(0.42408\) | \(\Gamma_0(N)\)-optimal |
900.b4 | 900e2 | \([0, 0, 0, 825, -9250]\) | \(21296/25\) | \(-72900000000\) | \([2]\) | \(576\) | \(0.77065\) | |
900.b1 | 900e3 | \([0, 0, 0, -9300, 345125]\) | \(488095744/125\) | \(22781250000\) | \([2]\) | \(864\) | \(0.97338\) | |
900.b2 | 900e4 | \([0, 0, 0, -8175, 431750]\) | \(-20720464/15625\) | \(-45562500000000\) | \([2]\) | \(1728\) | \(1.3200\) |
Rank
sage: E.rank()
The elliptic curves in class 900e have rank \(1\).
Complex multiplication
The elliptic curves in class 900e do not have complex multiplication.Modular form 900.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.