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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 10080.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10080.b1 | 10080n3 | \([0, 0, 0, -918540003, 10715075262502]\) | \(229625675762164624948320008/9568125\) | \(3571283520000\) | \([2]\) | \(1720320\) | \(3.3093\) | |
10080.b2 | 10080n2 | \([0, 0, 0, -57605628, 166216898752]\) | \(7079962908642659949376/100085966990454375\) | \(298855096058024916480000\) | \([2]\) | \(1720320\) | \(3.3093\) | |
10080.b3 | 10080n1 | \([0, 0, 0, -57408753, 167423033752]\) | \(448487713888272974160064/91549016015625\) | \(4271310891225000000\) | \([2, 2]\) | \(860160\) | \(2.9627\) | \(\Gamma_0(N)\)-optimal |
10080.b4 | 10080n4 | \([0, 0, 0, -57211923, 168628066378]\) | \(-55486311952875723077768/801237030029296875\) | \(-299060118984375000000000\) | \([2]\) | \(1720320\) | \(3.3093\) |
Rank
sage: E.rank()
The elliptic curves in class 10080.b have rank \(0\).
Complex multiplication
The elliptic curves in class 10080.b do not have complex multiplication.Modular form 10080.2.a.b
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.