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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 303240ce
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
303240.ce4 | 303240ce1 | \([0, 1, 0, -265455, -52730802]\) | \(2748251600896/2205\) | \(1659778681680\) | \([2]\) | \(1769472\) | \(1.6502\) | \(\Gamma_0(N)\)-optimal |
303240.ce3 | 303240ce2 | \([0, 1, 0, -267260, -51979200]\) | \(175293437776/4862025\) | \(58556991889670400\) | \([2, 2]\) | \(3538944\) | \(1.9968\) | |
303240.ce2 | 303240ce3 | \([0, 1, 0, -621040, 114155888]\) | \(549871953124/200930625\) | \(9679829271557760000\) | \([2, 2]\) | \(7077888\) | \(2.3434\) | |
303240.ce5 | 303240ce4 | \([0, 1, 0, 57640, -169982880]\) | \(439608956/259416045\) | \(-12497363335742100480\) | \([2]\) | \(7077888\) | \(2.3434\) | |
303240.ce1 | 303240ce5 | \([0, 1, 0, -8808520, 10057031600]\) | \(784478485879202/221484375\) | \(21340011621600000000\) | \([2]\) | \(14155776\) | \(2.6900\) | |
303240.ce6 | 303240ce6 | \([0, 1, 0, 1905960, 809586288]\) | \(7947184069438/7533176175\) | \(-725821255436462438400\) | \([2]\) | \(14155776\) | \(2.6900\) |
Rank
sage: E.rank()
The elliptic curves in class 303240ce have rank \(1\).
Complex multiplication
The elliptic curves in class 303240ce do not have complex multiplication.Modular form 303240.2.a.ce
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.