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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 195a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
195.a6 | 195a1 | \([1, 0, 0, -110, 435]\) | \(147281603041/5265\) | \(5265\) | \([4]\) | \(24\) | \(-0.19688\) | \(\Gamma_0(N)\)-optimal |
195.a5 | 195a2 | \([1, 0, 0, -115, 392]\) | \(168288035761/27720225\) | \(27720225\) | \([2, 4]\) | \(48\) | \(0.14970\) | |
195.a4 | 195a3 | \([1, 0, 0, -520, -4225]\) | \(15551989015681/1445900625\) | \(1445900625\) | \([2, 4]\) | \(96\) | \(0.49627\) | |
195.a7 | 195a4 | \([1, 0, 0, 210, 2277]\) | \(1023887723039/2798036865\) | \(-2798036865\) | \([4]\) | \(96\) | \(0.49627\) | |
195.a2 | 195a5 | \([1, 0, 0, -8125, -282568]\) | \(59319456301170001/594140625\) | \(594140625\) | \([2, 2]\) | \(192\) | \(0.84284\) | |
195.a8 | 195a6 | \([1, 0, 0, 605, -19750]\) | \(24487529386319/183539412225\) | \(-183539412225\) | \([4]\) | \(192\) | \(0.84284\) | |
195.a1 | 195a7 | \([1, 0, 0, -130000, -18051943]\) | \(242970740812818720001/24375\) | \(24375\) | \([2]\) | \(384\) | \(1.1894\) | |
195.a3 | 195a8 | \([1, 0, 0, -7930, -296725]\) | \(-55150149867714721/5950927734375\) | \(-5950927734375\) | \([2]\) | \(384\) | \(1.1894\) |
Rank
sage: E.rank()
The elliptic curves in class 195a have rank \(0\).
Complex multiplication
The elliptic curves in class 195a do not have complex multiplication.Modular form 195.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.