Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 6270.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6270.k1 | 6270g4 | \([1, 0, 1, -170319, -27068774]\) | \(546398303575251662569/180184125000\) | \(180184125000\) | \([2]\) | \(27648\) | \(1.5185\) | |
6270.k2 | 6270g3 | \([1, 0, 1, -10599, -427478]\) | \(-131661708271504489/2423495448000\) | \(-2423495448000\) | \([2]\) | \(13824\) | \(1.1719\) | |
6270.k3 | 6270g2 | \([1, 0, 1, -2454, -24098]\) | \(1633401545467609/698631332850\) | \(698631332850\) | \([6]\) | \(9216\) | \(0.96921\) | |
6270.k4 | 6270g1 | \([1, 0, 1, 516, -2714]\) | \(15236391945671/12100510620\) | \(-12100510620\) | \([6]\) | \(4608\) | \(0.62264\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6270.k have rank \(1\).
Complex multiplication
The elliptic curves in class 6270.k do not have complex multiplication.Modular form 6270.2.a.k
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.