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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 51714ba
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51714.s2 | 51714ba1 | \([1, -1, 1, 152375269, -1584967066549]\) | \(50611530622079699/169662750916608\) | \(-1311608442537419414528065536\) | \([2]\) | \(23482368\) | \(3.8865\) | \(\Gamma_0(N)\)-optimal |
51714.s1 | 51714ba2 | \([1, -1, 1, -1467428891, -18734805590965]\) | \(45204035637810785581/6545053349462016\) | \(50597713308513072449741340672\) | \([2]\) | \(46964736\) | \(4.2331\) |
Rank
sage: E.rank()
The elliptic curves in class 51714ba have rank \(0\).
Complex multiplication
The elliptic curves in class 51714ba do not have complex multiplication.Modular form 51714.2.a.ba
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.