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SageMath
E = EllipticCurve("gt1")
E.isogeny_class()
Elliptic curves in class 88200gt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 88200.hm4 | 88200gt1 | \([0, 0, 0, 11025, 2315250]\) | \(432/7\) | \(-2401451388000000\) | \([2]\) | \(393216\) | \(1.6318\) | \(\Gamma_0(N)\)-optimal |
| 88200.hm3 | 88200gt2 | \([0, 0, 0, -209475, 34728750]\) | \(740772/49\) | \(67240638864000000\) | \([2, 2]\) | \(786432\) | \(1.9784\) | |
| 88200.hm2 | 88200gt3 | \([0, 0, 0, -650475, -159752250]\) | \(11090466/2401\) | \(6589582608672000000\) | \([2]\) | \(1572864\) | \(2.3249\) | |
| 88200.hm1 | 88200gt4 | \([0, 0, 0, -3296475, 2303673750]\) | \(1443468546/7\) | \(19211611104000000\) | \([2]\) | \(1572864\) | \(2.3249\) |
Rank
sage: E.rank()
The elliptic curves in class 88200gt have rank \(0\).
Complex multiplication
The elliptic curves in class 88200gt do not have complex multiplication.Modular form 88200.2.a.gt
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.
