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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 78045a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 78045.c4 | 78045a1 | \([1, 1, 1, 179, -8326]\) | \(357911/17415\) | \(-30851734815\) | \([2]\) | \(56320\) | \(0.69244\) | \(\Gamma_0(N)\)-optimal |
| 78045.c3 | 78045a2 | \([1, 1, 1, -5266, -143362]\) | \(9116230969/416025\) | \(737013665025\) | \([2, 2]\) | \(112640\) | \(1.0390\) | |
| 78045.c2 | 78045a3 | \([1, 1, 1, -14341, 470108]\) | \(184122897769/51282015\) | \(90849217775415\) | \([2]\) | \(225280\) | \(1.3856\) | |
| 78045.c1 | 78045a4 | \([1, 1, 1, -83311, -9290236]\) | \(36097320816649/80625\) | \(142832105625\) | \([2]\) | \(225280\) | \(1.3856\) |
Rank
sage: E.rank()
The elliptic curves in class 78045a have rank \(2\).
Complex multiplication
The elliptic curves in class 78045a do not have complex multiplication.Modular form 78045.2.a.a
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.
