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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 42432bn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42432.bd4 | 42432bn1 | \([0, -1, 0, -157, -1955]\) | \(-420616192/1456611\) | \(-1491569664\) | \([2]\) | \(24576\) | \(0.44630\) | \(\Gamma_0(N)\)-optimal |
42432.bd3 | 42432bn2 | \([0, -1, 0, -3537, -79695]\) | \(298766385232/439569\) | \(7201898496\) | \([2, 2]\) | \(49152\) | \(0.79287\) | |
42432.bd2 | 42432bn3 | \([0, -1, 0, -4577, -27903]\) | \(161838334948/87947613\) | \(5763734765568\) | \([2]\) | \(98304\) | \(1.1394\) | |
42432.bd1 | 42432bn4 | \([0, -1, 0, -56577, -5160927]\) | \(305612563186948/663\) | \(43450368\) | \([2]\) | \(98304\) | \(1.1394\) |
Rank
sage: E.rank()
The elliptic curves in class 42432bn have rank \(1\).
Complex multiplication
The elliptic curves in class 42432bn do not have complex multiplication.Modular form 42432.2.a.bn
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.