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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 5415e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5415.f2 | 5415e1 | \([0, -1, 1, -215, 1256]\) | \(3058794496/91125\) | \(32896125\) | \([]\) | \(1296\) | \(0.21925\) | \(\Gamma_0(N)\)-optimal |
| 5415.f1 | 5415e2 | \([0, -1, 1, -17315, 882761]\) | \(1590409933520896/45\) | \(16245\) | \([]\) | \(3888\) | \(0.76855\) |
Rank
sage: E.rank()
The elliptic curves in class 5415e have rank \(1\).
Complex multiplication
The elliptic curves in class 5415e do not have complex multiplication.Modular form 5415.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
