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SageMath
E = EllipticCurve("dn1")
E.isogeny_class()
Elliptic curves in class 68544dn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68544.dw4 | 68544dn1 | \([0, 0, 0, 9321, -858688]\) | \(1919569026752/7938130977\) | \(-370361438862912\) | \([2]\) | \(229376\) | \(1.4781\) | \(\Gamma_0(N)\)-optimal |
| 68544.dw3 | 68544dn2 | \([0, 0, 0, -98724, -10539520]\) | \(35637273157312/4552605729\) | \(13594007865102336\) | \([2, 2]\) | \(458752\) | \(1.8246\) | |
| 68544.dw2 | 68544dn3 | \([0, 0, 0, -398604, 86021840]\) | \(293204888234504/35857918593\) | \(856569369536004096\) | \([2]\) | \(917504\) | \(2.1712\) | |
| 68544.dw1 | 68544dn4 | \([0, 0, 0, -1527564, -726674128]\) | \(16502300582616584/331494849\) | \(7918706521571328\) | \([2]\) | \(917504\) | \(2.1712\) |
Rank
sage: E.rank()
The elliptic curves in class 68544dn have rank \(1\).
Complex multiplication
The elliptic curves in class 68544dn do not have complex multiplication.Modular form 68544.2.a.dn
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.
