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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 333795.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333795.bc1 | 333795bc3 | \([1, 0, 0, -254615, 49429452]\) | \(75627935783569/396165\) | \(9562460022885\) | \([2]\) | \(1769472\) | \(1.6868\) | |
333795.bc2 | 333795bc2 | \([1, 0, 0, -16190, 743067]\) | \(19443408769/1334025\) | \(32200120485225\) | \([2, 2]\) | \(884736\) | \(1.3402\) | |
333795.bc3 | 333795bc1 | \([1, 0, 0, -3185, -55440]\) | \(148035889/31185\) | \(752730089265\) | \([2]\) | \(442368\) | \(0.99360\) | \(\Gamma_0(N)\)-optimal |
333795.bc4 | 333795bc4 | \([1, 0, 0, 14155, 3213150]\) | \(12994449551/192163125\) | \(-4638350688943125\) | \([2]\) | \(1769472\) | \(1.6868\) |
Rank
sage: E.rank()
The elliptic curves in class 333795.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 333795.bc do not have complex multiplication.Modular form 333795.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.