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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 16800e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 16800.b3 | 16800e1 | \([0, -1, 0, -506658, -138263688]\) | \(14383655824793536/45209390625\) | \(45209390625000000\) | \([2, 2]\) | \(184320\) | \(2.0630\) | \(\Gamma_0(N)\)-optimal |
| 16800.b1 | 16800e2 | \([0, -1, 0, -8100408, -8871076188]\) | \(7347751505995469192/72930375\) | \(583443000000000\) | \([2]\) | \(368640\) | \(2.4096\) | |
| 16800.b2 | 16800e3 | \([0, -1, 0, -725408, -7013688]\) | \(5276930158229192/3050936350875\) | \(24407490807000000000\) | \([2]\) | \(368640\) | \(2.4096\) | |
| 16800.b4 | 16800e4 | \([0, -1, 0, -294033, -255420063]\) | \(-43927191786304/415283203125\) | \(-26578125000000000000\) | \([2]\) | \(368640\) | \(2.4096\) |
Rank
sage: E.rank()
The elliptic curves in class 16800e have rank \(1\).
Complex multiplication
The elliptic curves in class 16800e do not have complex multiplication.Modular form 16800.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
