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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 20a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
20.a4 | 20a1 | \([0, 1, 0, 4, 4]\) | \(21296/25\) | \(-6400\) | \([6]\) | \(1\) | \(-0.58338\) | \(\Gamma_0(N)\)-optimal |
20.a3 | 20a2 | \([0, 1, 0, -1, 0]\) | \(16384/5\) | \(80\) | \([6]\) | \(2\) | \(-0.92995\) | |
20.a2 | 20a3 | \([0, 1, 0, -36, -140]\) | \(-20720464/15625\) | \(-4000000\) | \([2]\) | \(3\) | \(-0.034070\) | |
20.a1 | 20a4 | \([0, 1, 0, -41, -116]\) | \(488095744/125\) | \(2000\) | \([2]\) | \(6\) | \(-0.38064\) |
Rank
sage: E.rank()
The elliptic curves in class 20a have rank \(0\).
Complex multiplication
The elliptic curves in class 20a do not have complex multiplication.Modular form 20.2.a.a
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.