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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 390d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 390.d4 | 390d1 | \([1, 0, 1, 3997, 3998]\) | \(7064514799444439/4094064000000\) | \(-4094064000000\) | \([6]\) | \(720\) | \(1.1095\) | \(\Gamma_0(N)\)-optimal |
| 390.d3 | 390d2 | \([1, 0, 1, -16003, 27998]\) | \(453198971846635561/261896250564000\) | \(261896250564000\) | \([6]\) | \(1440\) | \(1.4561\) | |
| 390.d2 | 390d3 | \([1, 0, 1, -53378, -5124652]\) | \(-16818951115904497561/1592332281446400\) | \(-1592332281446400\) | \([2]\) | \(2160\) | \(1.6588\) | |
| 390.d1 | 390d4 | \([1, 0, 1, -872578, -313799212]\) | \(73474353581350183614361/576510977802240\) | \(576510977802240\) | \([2]\) | \(4320\) | \(2.0054\) |
Rank
sage: E.rank()
The elliptic curves in class 390d have rank \(0\).
Complex multiplication
The elliptic curves in class 390d do not have complex multiplication.Modular form 390.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
