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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 1323.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
1323.i1 | 1323b4 | \([0, 0, 1, -13230, 585758]\) | \(-12288000\) | \(-20841167403\) | \([]\) | \(1134\) | \(1.0251\) | \(-27\) | |
1323.i2 | 1323b2 | \([0, 0, 1, -1470, -21695]\) | \(-12288000\) | \(-28588707\) | \([]\) | \(378\) | \(0.47580\) | \(-27\) | |
1323.i3 | 1323b1 | \([0, 0, 1, 0, -86]\) | \(0\) | \(-3176523\) | \([]\) | \(126\) | \(-0.073509\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
1323.i4 | 1323b3 | \([0, 0, 1, 0, 2315]\) | \(0\) | \(-2315685267\) | \([]\) | \(378\) | \(0.47580\) | \(-3\) |
Rank
sage: E.rank()
The elliptic curves in class 1323.i have rank \(0\).
Complex multiplication
Each elliptic curve in class 1323.i has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 1323.2.a.i
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.
\(\left(\begin{array}{rrrr} 1 & 27 & 9 & 3 \\ 27 & 1 & 3 & 9 \\ 9 & 3 & 1 & 3 \\ 3 & 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.