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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 123627e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 123627.q5 | 123627e1 | \([1, 1, 0, -32308714, -75694250393]\) | \(-53297461115137/4513839183\) | \(-315880131146988948327207\) | \([2]\) | \(15482880\) | \(3.2541\) | \(\Gamma_0(N)\)-optimal |
| 123627.q4 | 123627e2 | \([1, 1, 0, -527022759, -4657043135520]\) | \(231331938231569617/1472026689\) | \(103012970715573655368681\) | \([2, 2]\) | \(30965760\) | \(3.6007\) | |
| 123627.q3 | 123627e3 | \([1, 1, 0, -537118964, -4469344588365]\) | \(244883173420511137/18418027974129\) | \(1288900391898782383598556441\) | \([2, 2]\) | \(61931520\) | \(3.9473\) | |
| 123627.q1 | 123627e4 | \([1, 1, 0, -8432351274, -298041176049903]\) | \(947531277805646290177/38367\) | \(2684936813291986743\) | \([2]\) | \(61931520\) | \(3.9473\) | |
| 123627.q6 | 123627e5 | \([1, 1, 0, 514328671, -19833307720512]\) | \(215015459663151503/2552757445339983\) | \(-178642907717531393896525690407\) | \([2]\) | \(123863040\) | \(4.2938\) | |
| 123627.q2 | 123627e6 | \([1, 1, 0, -1750105879, 22907527485802]\) | \(8471112631466271697/1662662681263647\) | \(116353747778338849791404863863\) | \([2]\) | \(123863040\) | \(4.2938\) |
Rank
sage: E.rank()
The elliptic curves in class 123627e have rank \(0\).
Complex multiplication
The elliptic curves in class 123627e do not have complex multiplication.Modular form 123627.2.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.
