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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 308763e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 308763.e5 | 308763e1 | \([1, -1, 1, -1192496, -536352150]\) | \(-53297461115137/4513839183\) | \(-15883043463338587263\) | \([2]\) | \(7077888\) | \(2.4293\) | \(\Gamma_0(N)\)-optimal |
| 308763.e4 | 308763e2 | \([1, -1, 1, -19452101, -33016537524]\) | \(231331938231569617/1472026689\) | \(5179684727944237329\) | \([2, 2]\) | \(14155776\) | \(2.7759\) | |
| 308763.e3 | 308763e3 | \([1, -1, 1, -19824746, -31685449584]\) | \(244883173420511137/18418027974129\) | \(64808321023889888159169\) | \([2, 2]\) | \(28311552\) | \(3.1224\) | |
| 308763.e1 | 308763e4 | \([1, -1, 1, -311233136, -2113298604660]\) | \(947531277805646290177/38367\) | \(135003641878287\) | \([2]\) | \(28311552\) | \(3.1224\) | |
| 308763.e2 | 308763e5 | \([1, -1, 1, -64595381, 162457932030]\) | \(8471112631466271697/1662662681263647\) | \(5850483936343989477356367\) | \([2]\) | \(56623104\) | \(3.4690\) | |
| 308763.e6 | 308763e6 | \([1, -1, 1, 18983569, -140643674778]\) | \(215015459663151503/2552757445339983\) | \(-8982499334136363705096063\) | \([2]\) | \(56623104\) | \(3.4690\) |
Rank
sage: E.rank()
The elliptic curves in class 308763e have rank \(0\).
Complex multiplication
The elliptic curves in class 308763e do not have complex multiplication.Modular form 308763.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
