Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 840b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
840.c4 | 840b1 | \([0, -1, 0, 9, -84]\) | \(4499456/180075\) | \(-2881200\) | \([4]\) | \(128\) | \(-0.080674\) | \(\Gamma_0(N)\)-optimal |
840.c3 | 840b2 | \([0, -1, 0, -236, -1260]\) | \(5702413264/275625\) | \(70560000\) | \([2, 2]\) | \(256\) | \(0.26590\) | |
840.c1 | 840b3 | \([0, -1, 0, -3736, -86660]\) | \(5633270409316/14175\) | \(14515200\) | \([2]\) | \(512\) | \(0.61247\) | |
840.c2 | 840b4 | \([0, -1, 0, -656, 4956]\) | \(30534944836/8203125\) | \(8400000000\) | \([2]\) | \(512\) | \(0.61247\) |
Rank
sage: E.rank()
The elliptic curves in class 840b have rank \(0\).
Complex multiplication
The elliptic curves in class 840b do not have complex multiplication.Modular form 840.2.a.b
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.