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SageMath
E = EllipticCurve("dp1")
E.isogeny_class()
Elliptic curves in class 485520.dp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485520.dp1 | 485520dp3 | \([0, -1, 0, -282993520, 1832462972032]\) | \(25351269426118370449/27551475\) | \(2723944975828070400\) | \([4]\) | \(56623104\) | \(3.2561\) | \(\Gamma_0(N)\)-optimal* |
485520.dp2 | 485520dp4 | \([0, -1, 0, -22061200, 13404640000]\) | \(12010404962647729/6166198828125\) | \(609636555495777600000000\) | \([2]\) | \(56623104\) | \(3.2561\) | |
485520.dp3 | 485520dp2 | \([0, -1, 0, -17691520, 28621613632]\) | \(6193921595708449/6452105625\) | \(637903440767900160000\) | \([2, 2]\) | \(28311552\) | \(2.9096\) | \(\Gamma_0(N)\)-optimal* |
485520.dp4 | 485520dp1 | \([0, -1, 0, -837040, 670144000]\) | \(-656008386769/1581036975\) | \(-156313145653714022400\) | \([2]\) | \(14155776\) | \(2.5630\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 485520.dp have rank \(0\).
Complex multiplication
The elliptic curves in class 485520.dp do not have complex multiplication.Modular form 485520.2.a.dp
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.