Show commands:
SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 3648w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3648.f1 | 3648w1 | \([0, -1, 0, -24, 54]\) | \(24897088/171\) | \(10944\) | \([2]\) | \(448\) | \(-0.39116\) | \(\Gamma_0(N)\)-optimal |
3648.f2 | 3648w2 | \([0, -1, 0, -9, 105]\) | \(-21952/1083\) | \(-4435968\) | \([2]\) | \(896\) | \(-0.044591\) |
Rank
sage: E.rank()
The elliptic curves in class 3648w have rank \(0\).
Complex multiplication
The elliptic curves in class 3648w do not have complex multiplication.Modular form 3648.2.a.w
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.