Show commands:
SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 9744g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
9744.o4 | 9744g1 | \([0, 1, 0, 41, 212]\) | \(464857088/1331883\) | \(-21310128\) | \([2]\) | \(2304\) | \(0.089691\) | \(\Gamma_0(N)\)-optimal |
9744.o3 | 9744g2 | \([0, 1, 0, -364, 2156]\) | \(20892021712/3337929\) | \(854509824\) | \([2, 2]\) | \(4608\) | \(0.43626\) | |
9744.o2 | 9744g3 | \([0, 1, 0, -1624, -23548]\) | \(462859546468/44558703\) | \(45628111872\) | \([2]\) | \(9216\) | \(0.78284\) | |
9744.o1 | 9744g4 | \([0, 1, 0, -5584, 158756]\) | \(18807789559108/626661\) | \(641700864\) | \([4]\) | \(9216\) | \(0.78284\) |
Rank
sage: E.rank()
The elliptic curves in class 9744g have rank \(0\).
Complex multiplication
The elliptic curves in class 9744g do not have complex multiplication.Modular form 9744.2.a.g
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.