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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 13680be
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
13680.y4 | 13680be1 | \([0, 0, 0, 1344237, -1275542062]\) | \(89962967236397039/287450726400000\) | \(-858323269818777600000\) | \([2]\) | \(460800\) | \(2.7000\) | \(\Gamma_0(N)\)-optimal |
13680.y3 | 13680be2 | \([0, 0, 0, -12664083, -14944860718]\) | \(75224183150104868881/11219310000000000\) | \(33500680151040000000000\) | \([2]\) | \(921600\) | \(3.0466\) | |
13680.y2 | 13680be3 | \([0, 0, 0, -475410963, -3989808404782]\) | \(-3979640234041473454886161/1471455901872240\) | \(-4393743779696078684160\) | \([2]\) | \(2304000\) | \(3.5047\) | |
13680.y1 | 13680be4 | \([0, 0, 0, -7606576083, -255347690321518]\) | \(16300610738133468173382620881/2228489100\) | \(6654232796774400\) | \([2]\) | \(4608000\) | \(3.8513\) |
Rank
sage: E.rank()
The elliptic curves in class 13680be have rank \(0\).
Complex multiplication
The elliptic curves in class 13680be do not have complex multiplication.Modular form 13680.2.a.be
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.