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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 448f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 448.a5 | 448f1 | \([0, 1, 0, -33, -161]\) | \(-15625/28\) | \(-7340032\) | \([2]\) | \(64\) | \(0.0076359\) | \(\Gamma_0(N)\)-optimal |
| 448.a4 | 448f2 | \([0, 1, 0, -673, -6945]\) | \(128787625/98\) | \(25690112\) | \([2]\) | \(128\) | \(0.35421\) | |
| 448.a6 | 448f3 | \([0, 1, 0, 287, 3231]\) | \(9938375/21952\) | \(-5754585088\) | \([2]\) | \(192\) | \(0.55694\) | |
| 448.a3 | 448f4 | \([0, 1, 0, -2273, 33439]\) | \(4956477625/941192\) | \(246727835648\) | \([2]\) | \(384\) | \(0.90352\) | |
| 448.a2 | 448f5 | \([0, 1, 0, -10913, 436447]\) | \(-548347731625/1835008\) | \(-481036337152\) | \([2]\) | \(576\) | \(1.1062\) | |
| 448.a1 | 448f6 | \([0, 1, 0, -174753, 28059871]\) | \(2251439055699625/25088\) | \(6576668672\) | \([2]\) | \(1152\) | \(1.4528\) |
Rank
sage: E.rank()
The elliptic curves in class 448f have rank \(0\).
Complex multiplication
The elliptic curves in class 448f do not have complex multiplication.Modular form 448.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
