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SageMath
E = EllipticCurve("et1")
E.isogeny_class()
Elliptic curves in class 68400et
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68400.l3 | 68400et1 | \([0, 0, 0, -28875, 144250]\) | \(57066625/32832\) | \(1531809792000000\) | \([2]\) | \(331776\) | \(1.6034\) | \(\Gamma_0(N)\)-optimal |
| 68400.l4 | 68400et2 | \([0, 0, 0, 115125, 1152250]\) | \(3616805375/2105352\) | \(-98227302912000000\) | \([2]\) | \(663552\) | \(1.9499\) | |
| 68400.l1 | 68400et3 | \([0, 0, 0, -1540875, -736199750]\) | \(8671983378625/82308\) | \(3840162048000000\) | \([2]\) | \(995328\) | \(2.1527\) | |
| 68400.l2 | 68400et4 | \([0, 0, 0, -1504875, -772235750]\) | \(-8078253774625/846825858\) | \(-39509507230848000000\) | \([2]\) | \(1990656\) | \(2.4992\) |
Rank
sage: E.rank()
The elliptic curves in class 68400et have rank \(1\).
Complex multiplication
The elliptic curves in class 68400et do not have complex multiplication.Modular form 68400.2.a.et
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
