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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 6762.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6762.bc1 | 6762bb4 | \([1, 1, 1, -8763602, 9981900641]\) | \(632678989847546725777/80515134\) | \(9472524999966\) | \([2]\) | \(184320\) | \(2.3499\) | |
6762.bc2 | 6762bb3 | \([1, 1, 1, -626662, 107865953]\) | \(231331938231569617/90942310746882\) | \(10699271917059920418\) | \([2]\) | \(184320\) | \(2.3499\) | |
6762.bc3 | 6762bb2 | \([1, 1, 1, -547772, 155767961]\) | \(154502321244119857/55101928644\) | \(6482686803037956\) | \([2, 2]\) | \(92160\) | \(2.0034\) | |
6762.bc4 | 6762bb1 | \([1, 1, 1, -29352, 3145113]\) | \(-23771111713777/22848457968\) | \(-2688098231477232\) | \([4]\) | \(46080\) | \(1.6568\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6762.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 6762.bc do not have complex multiplication.Modular form 6762.2.a.bc
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.