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SageMath
E = EllipticCurve("ea1")
E.isogeny_class()
Elliptic curves in class 72450.ea
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72450.ea1 | 72450eq4 | \([1, -1, 1, -40241030, 98264399847]\) | \(632678989847546725777/80515134\) | \(917117698218750\) | \([2]\) | \(3932160\) | \(2.7310\) | |
72450.ea2 | 72450eq3 | \([1, -1, 1, -2877530, 1064651847]\) | \(231331938231569617/90942310746882\) | \(1035889758351202781250\) | \([2]\) | \(3932160\) | \(2.7310\) | |
72450.ea3 | 72450eq2 | \([1, -1, 1, -2515280, 1535576847]\) | \(154502321244119857/55101928644\) | \(627645405960562500\) | \([2, 2]\) | \(1966080\) | \(2.3844\) | |
72450.ea4 | 72450eq1 | \([1, -1, 1, -134780, 31100847]\) | \(-23771111713777/22848457968\) | \(-260258216541750000\) | \([2]\) | \(983040\) | \(2.0379\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 72450.ea have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.ea do not have complex multiplication.Modular form 72450.2.a.ea
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.