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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 350350.ek
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 350350.ek1 | 350350ek4 | \([1, 0, 0, -96757799188, 11581508885806992]\) | \(54497099771831721530744218729/16209843781074944000000\) | \(29797998609370095104000000000000\) | \([2]\) | \(1672151040\) | \(5.0178\) | |
| 350350.ek2 | 350350ek3 | \([1, 0, 0, -6842407188, 130334137646992]\) | \(19272683606216463573689449/7161126378530668544000\) | \(13164052457933665992704000000000\) | \([2]\) | \(836075520\) | \(4.6712\) | |
| 350350.ek3 | 350350ek2 | \([1, 0, 0, -3225600813, -49947524526383]\) | \(2019051077229077416165369/582160888682835862400\) | \(1070166349885108693367150000000\) | \([2]\) | \(557383680\) | \(4.4685\) | |
| 350350.ek4 | 350350ek1 | \([1, 0, 0, -2956688813, -61873502814383]\) | \(1555006827939811751684089/221961497899581440\) | \(408024191662310263040000000\) | \([2]\) | \(278691840\) | \(4.1219\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 350350.ek have rank \(1\).
Complex multiplication
The elliptic curves in class 350350.ek do not have complex multiplication.Modular form 350350.2.a.ek
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.
