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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 9945k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 9945.h3 | 9945k1 | \([1, -1, 0, -234, -1265]\) | \(1948441249/89505\) | \(65249145\) | \([2]\) | \(3584\) | \(0.26111\) | \(\Gamma_0(N)\)-optimal |
| 9945.h2 | 9945k2 | \([1, -1, 0, -639, 4648]\) | \(39616946929/10989225\) | \(8011145025\) | \([2, 2]\) | \(7168\) | \(0.60769\) | |
| 9945.h1 | 9945k3 | \([1, -1, 0, -9414, 353893]\) | \(126574061279329/16286595\) | \(11872927755\) | \([4]\) | \(14336\) | \(0.95426\) | |
| 9945.h4 | 9945k4 | \([1, -1, 0, 1656, 28975]\) | \(688699320191/910381875\) | \(-663668386875\) | \([2]\) | \(14336\) | \(0.95426\) |
Rank
sage: E.rank()
The elliptic curves in class 9945k have rank \(1\).
Complex multiplication
The elliptic curves in class 9945k do not have complex multiplication.Modular form 9945.2.a.k
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.
