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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 159936o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
159936.go4 | 159936o1 | \([0, 1, 0, 3071, 17697791]\) | \(103823/4386816\) | \(-135293702133252096\) | \([2]\) | \(1769472\) | \(1.9661\) | \(\Gamma_0(N)\)-optimal |
159936.go3 | 159936o2 | \([0, 1, 0, -1000449, 377961471]\) | \(3590714269297/73410624\) | \(2264055546636140544\) | \([2, 2]\) | \(3538944\) | \(2.3127\) | |
159936.go1 | 159936o3 | \([0, 1, 0, -15927809, 24461764095]\) | \(14489843500598257/6246072\) | \(192634978232696832\) | \([2]\) | \(7077888\) | \(2.6592\) | |
159936.go2 | 159936o4 | \([0, 1, 0, -2129409, -632909313]\) | \(34623662831857/14438442312\) | \(445295702721033142272\) | \([2]\) | \(7077888\) | \(2.6592\) |
Rank
sage: E.rank()
The elliptic curves in class 159936o have rank \(0\).
Complex multiplication
The elliptic curves in class 159936o do not have complex multiplication.Modular form 159936.2.a.o
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.