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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 858b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 858.c4 | 858b1 | \([1, 0, 1, 359, 1916]\) | \(5137417856375/4510142208\) | \(-4510142208\) | \([6]\) | \(576\) | \(0.53994\) | \(\Gamma_0(N)\)-optimal |
| 858.c3 | 858b2 | \([1, 0, 1, -1801, 16604]\) | \(645532578015625/252306960048\) | \(252306960048\) | \([6]\) | \(1152\) | \(0.88652\) | |
| 858.c2 | 858b3 | \([1, 0, 1, -3736, -117658]\) | \(-5764706497797625/2612665516032\) | \(-2612665516032\) | \([2]\) | \(1728\) | \(1.0893\) | |
| 858.c1 | 858b4 | \([1, 0, 1, -65176, -6409114]\) | \(30618029936661765625/3678951124992\) | \(3678951124992\) | \([2]\) | \(3456\) | \(1.4358\) |
Rank
sage: E.rank()
The elliptic curves in class 858b have rank \(1\).
Complex multiplication
The elliptic curves in class 858b do not have complex multiplication.Modular form 858.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
