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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 388080z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
388080.z3 | 388080z1 | \([0, 0, 0, -6445803, -6325560038]\) | \(-84309998289049/414124480\) | \(-145481114667999559680\) | \([2]\) | \(15925248\) | \(2.7170\) | \(\Gamma_0(N)\)-optimal |
388080.z2 | 388080z2 | \([0, 0, 0, -103254123, -403839883622]\) | \(346553430870203929/8300600\) | \(2915984441231769600\) | \([2]\) | \(31850496\) | \(3.0636\) | |
388080.z4 | 388080z3 | \([0, 0, 0, 16027557, -33584362742]\) | \(1296134247276791/2137096192000\) | \(-750757685623649206272000\) | \([2]\) | \(47775744\) | \(3.2664\) | |
388080.z1 | 388080z4 | \([0, 0, 0, -110415963, -344610133238]\) | \(423783056881319689/99207416000000\) | \(34851369962509664256000000\) | \([2]\) | \(95551488\) | \(3.6129\) |
Rank
sage: E.rank()
The elliptic curves in class 388080z have rank \(0\).
Complex multiplication
The elliptic curves in class 388080z do not have complex multiplication.Modular form 388080.2.a.z
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.