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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 304200.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 304200.k1 | 304200k3 | \([0, 0, 0, -62982075, 191659139750]\) | \(490757540836/2142075\) | \(120598608669505200000000\) | \([2]\) | \(49545216\) | \(3.2818\) | |
| 304200.k2 | 304200k2 | \([0, 0, 0, -5944575, -386122750]\) | \(1650587344/950625\) | \(13380023151202500000000\) | \([2, 2]\) | \(24772608\) | \(2.9352\) | |
| 304200.k3 | 304200k1 | \([0, 0, 0, -4233450, -3344657875]\) | \(9538484224/26325\) | \(23157732377081250000\) | \([2]\) | \(12386304\) | \(2.5887\) | \(\Gamma_0(N)\)-optimal |
| 304200.k4 | 304200k4 | \([0, 0, 0, 23714925, -3085137250]\) | \(26198797244/15234375\) | \(-857693791743750000000000\) | \([2]\) | \(49545216\) | \(3.2818\) |
Rank
sage: E.rank()
The elliptic curves in class 304200.k have rank \(1\).
Complex multiplication
The elliptic curves in class 304200.k do not have complex multiplication.Modular form 304200.2.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 LMFDB 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 LMFDB labels.
