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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 103155h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 103155.e6 | 103155h1 | \([1, 0, 0, -58201, -5409040]\) | \(147281603041/5265\) | \(779408955585\) | \([2]\) | \(270336\) | \(1.3709\) | \(\Gamma_0(N)\)-optimal |
| 103155.e5 | 103155h2 | \([1, 0, 0, -60846, -4891149]\) | \(168288035761/27720225\) | \(4103588151155025\) | \([2, 2]\) | \(540672\) | \(1.7174\) | |
| 103155.e7 | 103155h3 | \([1, 0, 0, 111079, -27482094]\) | \(1023887723039/2798036865\) | \(-414209874765047985\) | \([2]\) | \(1081344\) | \(2.0640\) | |
| 103155.e4 | 103155h4 | \([1, 0, 0, -275091, 50855400]\) | \(15551989015681/1445900625\) | \(214045184427530625\) | \([2, 2]\) | \(1081344\) | \(2.0640\) | |
| 103155.e8 | 103155h5 | \([1, 0, 0, 320034, 240938325]\) | \(24487529386319/183539412225\) | \(-27170420055265343025\) | \([2]\) | \(2162688\) | \(2.4106\) | |
| 103155.e2 | 103155h6 | \([1, 0, 0, -4298136, 3429408591]\) | \(59319456301170001/594140625\) | \(87954135612890625\) | \([2, 2]\) | \(2162688\) | \(2.4106\) | |
| 103155.e3 | 103155h7 | \([1, 0, 0, -4194981, 3601863120]\) | \(-55150149867714721/5950927734375\) | \(-880950877532958984375\) | \([2]\) | \(4325376\) | \(2.7572\) | |
| 103155.e1 | 103155h8 | \([1, 0, 0, -68770011, 219500450466]\) | \(242970740812818720001/24375\) | \(3608374794375\) | \([2]\) | \(4325376\) | \(2.7572\) |
Rank
sage: E.rank()
The elliptic curves in class 103155h have rank \(2\).
Complex multiplication
The elliptic curves in class 103155h do not have complex multiplication.Modular form 103155.2.a.h
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
