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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 320320.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 320320.bc1 | 320320bc4 | \([0, 1, 0, -5055101345, 138301211242975]\) | \(54497099771831721530744218729/16209843781074944000000\) | \(4249313288146110119936000000\) | \([2]\) | \(278691840\) | \(4.2798\) | |
| 320320.bc2 | 320320bc3 | \([0, 1, 0, -357480865, 1556297642463]\) | \(19272683606216463573689449/7161126378530668544000\) | \(1877246313373543574798336000\) | \([2]\) | \(139345920\) | \(3.9332\) | |
| 320320.bc3 | 320320bc2 | \([0, 1, 0, -168521185, -596510866017]\) | \(2019051077229077416165369/582160888682835862400\) | \(152609984002873324312985600\) | \([2]\) | \(92897280\) | \(3.7305\) | |
| 320320.bc4 | 320320bc1 | \([0, 1, 0, -154471905, -738922797665]\) | \(1555006827939811751684089/221961497899581440\) | \(58185874905387877007360\) | \([2]\) | \(46448640\) | \(3.3839\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 320320.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 320320.bc do not have complex multiplication.Modular form 320320.2.a.bc
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.
