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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 49686k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
49686.u3 | 49686k1 | \([1, 1, 0, -58139, 4810317]\) | \(38272753/4368\) | \(2480452892915088\) | \([2]\) | \(387072\) | \(1.6863\) | \(\Gamma_0(N)\)-optimal |
49686.u2 | 49686k2 | \([1, 1, 0, -223759, -35700335]\) | \(2181825073/298116\) | \(169290909941454756\) | \([2, 2]\) | \(774144\) | \(2.0328\) | |
49686.u4 | 49686k3 | \([1, 1, 0, 355911, -189312885]\) | \(8780064047/32388174\) | \(-18392248144353763134\) | \([2]\) | \(1548288\) | \(2.3794\) | |
49686.u1 | 49686k4 | \([1, 1, 0, -3453349, -2471457113]\) | \(8020417344913/187278\) | \(106349417783734398\) | \([2]\) | \(1548288\) | \(2.3794\) |
Rank
sage: E.rank()
The elliptic curves in class 49686k have rank \(0\).
Complex multiplication
The elliptic curves in class 49686k do not have complex multiplication.Modular form 49686.2.a.k
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.