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SageMath
E = EllipticCurve("el1")
E.isogeny_class()
Elliptic curves in class 72450.el
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72450.el1 | 72450ek4 | \([1, -1, 1, -40572005, -99458871253]\) | \(648418741232906810881/33810\) | \(385117031250\) | \([2]\) | \(2359296\) | \(2.6164\) | |
72450.el2 | 72450ek3 | \([1, -1, 1, -2587505, -1486806253]\) | \(168197522113656001/13424780328750\) | \(152916638432167968750\) | \([2]\) | \(2359296\) | \(2.6164\) | |
72450.el3 | 72450ek2 | \([1, -1, 1, -2535755, -1553563753]\) | \(158306179791523681/1143116100\) | \(13020806826562500\) | \([2, 2]\) | \(1179648\) | \(2.2699\) | |
72450.el4 | 72450ek1 | \([1, -1, 1, -155255, -25282753]\) | \(-36333758230561/3290930160\) | \(-37485751353750000\) | \([4]\) | \(589824\) | \(1.9233\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 72450.el have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.el do not have complex multiplication.Modular form 72450.2.a.el
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.