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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 1050h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1050.i5 | 1050h1 | \([1, 0, 1, -101, 848]\) | \(-7189057/16128\) | \(-252000000\) | \([2]\) | \(512\) | \(0.30067\) | \(\Gamma_0(N)\)-optimal |
| 1050.i4 | 1050h2 | \([1, 0, 1, -2101, 36848]\) | \(65597103937/63504\) | \(992250000\) | \([2, 2]\) | \(1024\) | \(0.64725\) | |
| 1050.i3 | 1050h3 | \([1, 0, 1, -2601, 17848]\) | \(124475734657/63011844\) | \(984560062500\) | \([2, 2]\) | \(2048\) | \(0.99382\) | |
| 1050.i1 | 1050h4 | \([1, 0, 1, -33601, 2367848]\) | \(268498407453697/252\) | \(3937500\) | \([2]\) | \(2048\) | \(0.99382\) | |
| 1050.i2 | 1050h5 | \([1, 0, 1, -22851, -1318652]\) | \(84448510979617/933897762\) | \(14592152531250\) | \([2]\) | \(4096\) | \(1.3404\) | |
| 1050.i6 | 1050h6 | \([1, 0, 1, 9649, 140348]\) | \(6359387729183/4218578658\) | \(-65915291531250\) | \([2]\) | \(4096\) | \(1.3404\) |
Rank
sage: E.rank()
The elliptic curves in class 1050h have rank \(1\).
Complex multiplication
The elliptic curves in class 1050h do not have complex multiplication.Modular form 1050.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
