Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 75843a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
75843.d4 | 75843a1 | \([0, 0, 1, 0, 37219]\) | \(0\) | \(-598437750483\) | \([]\) | \(50544\) | \(0.93868\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
75843.d2 | 75843a2 | \([0, 0, 1, -84270, 9416470]\) | \(-12288000\) | \(-5385939754347\) | \([]\) | \(151632\) | \(1.4880\) | \(-27\) | |
75843.d3 | 75843a3 | \([0, 0, 1, 0, -1004920]\) | \(0\) | \(-436261120102107\) | \([]\) | \(151632\) | \(1.4880\) | \(-3\) | |
75843.d1 | 75843a4 | \([0, 0, 1, -758430, -254244697]\) | \(-12288000\) | \(-3926350080918963\) | \([]\) | \(454896\) | \(2.0373\) | \(-27\) |
Rank
sage: E.rank()
The elliptic curves in class 75843a have rank \(1\).
Complex multiplication
Each elliptic curve in class 75843a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 75843.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 3 & 9 \\ 3 & 1 & 9 & 27 \\ 3 & 9 & 1 & 3 \\ 9 & 27 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.