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SageMath
E = EllipticCurve("dw1")
E.isogeny_class()
Elliptic curves in class 107712dw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 107712.y4 | 107712dw1 | \([0, 0, 0, -3839916, 2178851024]\) | \(32765849647039657/8229948198912\) | \(1572767593137923162112\) | \([2]\) | \(4128768\) | \(2.7772\) | \(\Gamma_0(N)\)-optimal |
| 107712.y2 | 107712dw2 | \([0, 0, 0, -57108396, 166096617680]\) | \(107784459654566688937/10704361149504\) | \(2045635271848995323904\) | \([2, 2]\) | \(8257536\) | \(3.1238\) | |
| 107712.y3 | 107712dw3 | \([0, 0, 0, -52799916, 192214623440]\) | \(-85183593440646799657/34223681512621656\) | \(-6540247386738180023648256\) | \([2]\) | \(16515072\) | \(3.4704\) | |
| 107712.y1 | 107712dw4 | \([0, 0, 0, -913712556, 10630715677904]\) | \(441453577446719855661097/4354701912\) | \(832196494976090112\) | \([2]\) | \(16515072\) | \(3.4704\) |
Rank
sage: E.rank()
The elliptic curves in class 107712dw have rank \(2\).
Complex multiplication
The elliptic curves in class 107712dw do not have complex multiplication.Modular form 107712.2.a.dw
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.
