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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 5415.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5415.j1 | 5415k7 | \([1, 0, 1, -779768, 264965501]\) | \(1114544804970241/405\) | \(19053581805\) | \([2]\) | \(27648\) | \(1.7631\) | |
5415.j2 | 5415k5 | \([1, 0, 1, -48743, 4135781]\) | \(272223782641/164025\) | \(7716700631025\) | \([2, 2]\) | \(13824\) | \(1.4165\) | |
5415.j3 | 5415k8 | \([1, 0, 1, -39718, 5716961]\) | \(-147281603041/215233605\) | \(-10125854568031005\) | \([2]\) | \(27648\) | \(1.7631\) | |
5415.j4 | 5415k3 | \([1, 0, 1, -28888, -1892197]\) | \(56667352321/15\) | \(705688215\) | \([2]\) | \(6912\) | \(1.0699\) | |
5415.j5 | 5415k4 | \([1, 0, 1, -3618, 38431]\) | \(111284641/50625\) | \(2381697725625\) | \([2, 2]\) | \(6912\) | \(1.0699\) | |
5415.j6 | 5415k2 | \([1, 0, 1, -1813, -29437]\) | \(13997521/225\) | \(10585323225\) | \([2, 2]\) | \(3456\) | \(0.72337\) | |
5415.j7 | 5415k1 | \([1, 0, 1, -8, -1279]\) | \(-1/15\) | \(-705688215\) | \([2]\) | \(1728\) | \(0.37679\) | \(\Gamma_0(N)\)-optimal |
5415.j8 | 5415k6 | \([1, 0, 1, 12627, 291853]\) | \(4733169839/3515625\) | \(-165395675390625\) | \([2]\) | \(13824\) | \(1.4165\) |
Rank
sage: E.rank()
The elliptic curves in class 5415.j have rank \(0\).
Complex multiplication
The elliptic curves in class 5415.j do not have complex multiplication.Modular form 5415.2.a.j
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.