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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 24990e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
24990.d3 | 24990e1 | \([1, 1, 0, -881878, -319273772]\) | \(-644706081631626841/347004000000\) | \(-40824673596000000\) | \([2]\) | \(442368\) | \(2.1365\) | \(\Gamma_0(N)\)-optimal |
24990.d2 | 24990e2 | \([1, 1, 0, -14111878, -20410351772]\) | \(2641739317048851306841/764694000\) | \(89965484406000\) | \([2]\) | \(884736\) | \(2.4831\) | |
24990.d4 | 24990e3 | \([1, 1, 0, 716747, -1295019347]\) | \(346124368852751159/6361262220902400\) | \(-748396139026946457600\) | \([2]\) | \(1327104\) | \(2.6858\) | |
24990.d1 | 24990e4 | \([1, 1, 0, -14336053, -19728678227]\) | \(2769646315294225853641/174474906948464640\) | \(20526798327579916431360\) | \([2]\) | \(2654208\) | \(3.0324\) |
Rank
sage: E.rank()
The elliptic curves in class 24990e have rank \(1\).
Complex multiplication
The elliptic curves in class 24990e do not have complex multiplication.Modular form 24990.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.