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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 37485.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
37485.j1 | 37485v4 | \([1, -1, 1, -9802778, -11810847324]\) | \(1214661886599131209/2213451765\) | \(189839171904653565\) | \([2]\) | \(1179648\) | \(2.5725\) | |
37485.j2 | 37485v3 | \([1, -1, 1, -1631048, 561211872]\) | \(5595100866606889/1653777286875\) | \(141838062893172961875\) | \([2]\) | \(1179648\) | \(2.5725\) | |
37485.j3 | 37485v2 | \([1, -1, 1, -618953, -180451344]\) | \(305759741604409/12646127025\) | \(1084609260607520025\) | \([2, 2]\) | \(589824\) | \(2.2259\) | |
37485.j4 | 37485v1 | \([1, -1, 1, 18292, -10434378]\) | \(7892485271/552491415\) | \(-47385045550351215\) | \([2]\) | \(294912\) | \(1.8793\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 37485.j have rank \(1\).
Complex multiplication
The elliptic curves in class 37485.j do not have complex multiplication.Modular form 37485.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.