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SageMath
E = EllipticCurve("bal1")
E.isogeny_class()
Elliptic curves in class 235200.bal
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 235200.bal1 | 235200bal3 | \([0, 1, 0, -5489633, 4948828863]\) | \(303735479048/105\) | \(6324810240000000\) | \([2]\) | \(7077888\) | \(2.3878\) | |
| 235200.bal2 | 235200bal2 | \([0, 1, 0, -344633, 76513863]\) | \(601211584/11025\) | \(83013134400000000\) | \([2, 2]\) | \(3538944\) | \(2.0413\) | |
| 235200.bal3 | 235200bal1 | \([0, 1, 0, -44508, -1818762]\) | \(82881856/36015\) | \(4237128735000000\) | \([2]\) | \(1769472\) | \(1.6947\) | \(\Gamma_0(N)\)-optimal |
| 235200.bal4 | 235200bal4 | \([0, 1, 0, -1633, 222288863]\) | \(-8/354375\) | \(-21346234560000000000\) | \([2]\) | \(7077888\) | \(2.3878\) |
Rank
sage: E.rank()
The elliptic curves in class 235200.bal have rank \(2\).
Complex multiplication
The elliptic curves in class 235200.bal do not have complex multiplication.Modular form 235200.2.a.bal
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
