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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 73689.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 73689.e1 | 73689s4 | \([1, 1, 1, -1442867, -667692376]\) | \(187519537050946633/1186707753\) | \(2102325173612433\) | \([2]\) | \(737280\) | \(2.1254\) | |
| 73689.e2 | 73689s2 | \([1, 1, 1, -91902, -10042614]\) | \(48455467135993/3635004681\) | \(6439632527677041\) | \([2, 2]\) | \(368640\) | \(1.7788\) | |
| 73689.e3 | 73689s1 | \([1, 1, 1, -18697, 791726]\) | \(408023180713/80247321\) | \(142163024238081\) | \([2]\) | \(184320\) | \(1.4322\) | \(\Gamma_0(N)\)-optimal |
| 73689.e4 | 73689s3 | \([1, 1, 1, 87783, -44326512]\) | \(42227808999767/504359959257\) | \(-893504433781290177\) | \([4]\) | \(737280\) | \(2.1254\) |
Rank
sage: E.rank()
The elliptic curves in class 73689.e have rank \(0\).
Complex multiplication
The elliptic curves in class 73689.e do not have complex multiplication.Modular form 73689.2.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 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.
