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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 193200.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 193200.f1 | 193200ek4 | \([0, -1, 0, -2020408, -1104676688]\) | \(14251520160844849/264449745\) | \(16924783680000000\) | \([2]\) | \(2949120\) | \(2.2389\) | |
| 193200.f2 | 193200ek2 | \([0, -1, 0, -130408, -16036688]\) | \(3832302404449/472410225\) | \(30234254400000000\) | \([2, 2]\) | \(1474560\) | \(1.8923\) | |
| 193200.f3 | 193200ek1 | \([0, -1, 0, -32408, 1995312]\) | \(58818484369/7455105\) | \(477126720000000\) | \([2]\) | \(737280\) | \(1.5457\) | \(\Gamma_0(N)\)-optimal |
| 193200.f4 | 193200ek3 | \([0, -1, 0, 191592, -83012688]\) | \(12152722588271/53476250625\) | \(-3422480040000000000\) | \([2]\) | \(2949120\) | \(2.2389\) |
Rank
sage: E.rank()
The elliptic curves in class 193200.f have rank \(0\).
Complex multiplication
The elliptic curves in class 193200.f do not have complex multiplication.Modular form 193200.2.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 LMFDB 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 LMFDB labels.
