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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 1110c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1110.a4 | 1110c1 | \([1, 1, 0, -1423388, -615252528]\) | \(318929057401476905525449/21353131537921474560\) | \(21353131537921474560\) | \([2]\) | \(49280\) | \(2.4580\) | \(\Gamma_0(N)\)-optimal |
1110.a2 | 1110c2 | \([1, 1, 0, -22394908, -40800879152]\) | \(1242142983306846366056931529/6179359141291622400\) | \(6179359141291622400\) | \([2, 2]\) | \(98560\) | \(2.8046\) | |
1110.a1 | 1110c3 | \([1, 1, 0, -358318108, -2610814913072]\) | \(5087799435928552778197163696329/125914832087040\) | \(125914832087040\) | \([2]\) | \(197120\) | \(3.1512\) | |
1110.a3 | 1110c4 | \([1, 1, 0, -22016028, -42247518768]\) | \(-1180159344892952613848670409/87759036144023189760000\) | \(-87759036144023189760000\) | \([2]\) | \(197120\) | \(3.1512\) |
Rank
sage: E.rank()
The elliptic curves in class 1110c have rank \(1\).
Complex multiplication
The elliptic curves in class 1110c do not have complex multiplication.Modular form 1110.2.a.c
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.