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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 3360.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3360.e1 | 3360n3 | \([0, -1, 0, -2416, 46516]\) | \(3047363673992/540225\) | \(276595200\) | \([4]\) | \(2048\) | \(0.62447\) | |
3360.e2 | 3360n2 | \([0, -1, 0, -1041, -12159]\) | \(30488290624/1148175\) | \(4702924800\) | \([2]\) | \(2048\) | \(0.62447\) | |
3360.e3 | 3360n1 | \([0, -1, 0, -166, 616]\) | \(7952095936/2480625\) | \(158760000\) | \([2, 2]\) | \(1024\) | \(0.27790\) | \(\Gamma_0(N)\)-optimal |
3360.e4 | 3360n4 | \([0, -1, 0, 464, 3640]\) | \(21531355768/24609375\) | \(-12600000000\) | \([2]\) | \(2048\) | \(0.62447\) |
Rank
sage: E.rank()
The elliptic curves in class 3360.e have rank \(1\).
Complex multiplication
The elliptic curves in class 3360.e do not have complex multiplication.Modular form 3360.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.