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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 3225e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3225.d4 | 3225e1 | \([1, 0, 0, 37, 792]\) | \(357911/17415\) | \(-272109375\) | \([4]\) | \(1056\) | \(0.29822\) | \(\Gamma_0(N)\)-optimal |
| 3225.d3 | 3225e2 | \([1, 0, 0, -1088, 13167]\) | \(9116230969/416025\) | \(6500390625\) | \([2, 2]\) | \(2112\) | \(0.64479\) | |
| 3225.d2 | 3225e3 | \([1, 0, 0, -2963, -44958]\) | \(184122897769/51282015\) | \(801281484375\) | \([2]\) | \(4224\) | \(0.99136\) | |
| 3225.d1 | 3225e4 | \([1, 0, 0, -17213, 867792]\) | \(36097320816649/80625\) | \(1259765625\) | \([2]\) | \(4224\) | \(0.99136\) |
Rank
sage: E.rank()
The elliptic curves in class 3225e have rank \(0\).
Complex multiplication
The elliptic curves in class 3225e do not have complex multiplication.Modular form 3225.2.a.e
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.
