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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 53900.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53900.e1 | 53900v4 | \([0, 1, 0, -240271908, 1431989153188]\) | \(3259751350395879376/3806353980275\) | \(1791254957701493900000000\) | \([2]\) | \(11943936\) | \(3.5659\) | |
53900.e2 | 53900v3 | \([0, 1, 0, -240204533, 1432833227188]\) | \(52112158467655991296/71177645\) | \(2093494689151250000\) | \([2]\) | \(5971968\) | \(3.2193\) | |
53900.e3 | 53900v2 | \([0, 1, 0, -11196908, -12396096812]\) | \(329890530231376/49933296875\) | \(23498409776187500000000\) | \([2]\) | \(3981312\) | \(3.0166\) | |
53900.e4 | 53900v1 | \([0, 1, 0, -3044533, 1854254688]\) | \(106110329552896/10850811125\) | \(319146769511281250000\) | \([2]\) | \(1990656\) | \(2.6700\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 53900.e have rank \(0\).
Complex multiplication
The elliptic curves in class 53900.e do not have complex multiplication.Modular form 53900.2.a.e
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 LMFDB 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 LMFDB labels.