Show commands:
SageMath
E = EllipticCurve("dv1")
E.isogeny_class()
Elliptic curves in class 486720dv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
486720.dv6 | 486720dv1 | \([0, 0, 0, 1458132, -271408592]\) | \(371694959/249600\) | \(-230235424095102566400\) | \([2]\) | \(16515072\) | \(2.5959\) | \(\Gamma_0(N)\)-optimal* |
486720.dv5 | 486720dv2 | \([0, 0, 0, -6329388, -2255668688]\) | \(30400540561/15210000\) | \(14029971155795312640000\) | \([2, 2]\) | \(33030144\) | \(2.9424\) | \(\Gamma_0(N)\)-optimal* |
486720.dv3 | 486720dv3 | \([0, 0, 0, -55001388, 155422142512]\) | \(19948814692561/231344100\) | \(213395861279646705254400\) | \([2, 2]\) | \(66060288\) | \(3.2890\) | \(\Gamma_0(N)\)-optimal* |
486720.dv2 | 486720dv4 | \([0, 0, 0, -82257708, -286926126032]\) | \(66730743078481/60937500\) | \(56209820335718400000000\) | \([2]\) | \(66060288\) | \(3.2890\) | |
486720.dv1 | 486720dv5 | \([0, 0, 0, -877558188, 10006033356592]\) | \(81025909800741361/11088090\) | \(10227848972574782914560\) | \([2]\) | \(132120576\) | \(3.6356\) | \(\Gamma_0(N)\)-optimal* |
486720.dv4 | 486720dv6 | \([0, 0, 0, -11196588, 396190845232]\) | \(-168288035761/73415764890\) | \(-67719991044533217208565760\) | \([2]\) | \(132120576\) | \(3.6356\) |
Rank
sage: E.rank()
The elliptic curves in class 486720dv have rank \(0\).
Complex multiplication
The elliptic curves in class 486720dv do not have complex multiplication.Modular form 486720.2.a.dv
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.