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SageMath
E = EllipticCurve("dj1")
E.isogeny_class()
Elliptic curves in class 64680dj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
64680.dl4 | 64680dj1 | \([0, 1, 0, 17820, -871200]\) | \(20777545136/23059575\) | \(-694511600428800\) | \([4]\) | \(196608\) | \(1.5344\) | \(\Gamma_0(N)\)-optimal |
64680.dl3 | 64680dj2 | \([0, 1, 0, -100760, -8270592]\) | \(939083699236/300155625\) | \(36160521344640000\) | \([2, 2]\) | \(393216\) | \(1.8809\) | |
64680.dl2 | 64680dj3 | \([0, 1, 0, -639760, 190512608]\) | \(120186986927618/4332064275\) | \(1043789885213644800\) | \([2]\) | \(786432\) | \(2.2275\) | |
64680.dl1 | 64680dj4 | \([0, 1, 0, -1459040, -678717600]\) | \(1425631925916578/270703125\) | \(65224605600000000\) | \([2]\) | \(786432\) | \(2.2275\) |
Rank
sage: E.rank()
The elliptic curves in class 64680dj have rank \(0\).
Complex multiplication
The elliptic curves in class 64680dj do not have complex multiplication.Modular form 64680.2.a.dj
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.