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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 41650p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
41650.s3 | 41650p1 | \([1, -1, 0, -22892, -1305984]\) | \(721734273/13328\) | \(24500404250000\) | \([2]\) | \(98304\) | \(1.3637\) | \(\Gamma_0(N)\)-optimal |
41650.s2 | 41650p2 | \([1, -1, 0, -47392, 2001516]\) | \(6403769793/2775556\) | \(5102209185062500\) | \([2, 2]\) | \(196608\) | \(1.7102\) | |
41650.s4 | 41650p3 | \([1, -1, 0, 160858, 14704766]\) | \(250404380127/196003234\) | \(-360306007451031250\) | \([2]\) | \(393216\) | \(2.0568\) | |
41650.s1 | 41650p4 | \([1, -1, 0, -647642, 200684266]\) | \(16342588257633/8185058\) | \(15046310760031250\) | \([2]\) | \(393216\) | \(2.0568\) |
Rank
sage: E.rank()
The elliptic curves in class 41650p have rank \(1\).
Complex multiplication
The elliptic curves in class 41650p do not have complex multiplication.Modular form 41650.2.a.p
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.