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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 8470.ba
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8470.ba1 | 8470be4 | \([1, -1, 1, -395266972, 3024809938571]\) | \(3855131356812007128171561/8967612500\) | \(15886672568112500\) | \([2]\) | \(1228800\) | \(3.2390\) | |
8470.ba2 | 8470be3 | \([1, -1, 1, -26008852, 41998530379]\) | \(1098325674097093229481/205612182617187500\) | \(364254523849487304687500\) | \([2]\) | \(1228800\) | \(3.2390\) | |
8470.ba3 | 8470be2 | \([1, -1, 1, -24704472, 47266138571]\) | \(941226862950447171561/45393906250000\) | \(80418073950156250000\) | \([2, 2]\) | \(614400\) | \(2.8924\) | |
8470.ba4 | 8470be1 | \([1, -1, 1, -1462792, 819965259]\) | \(-195395722614328041/50730248800000\) | \(-89871730294376800000\) | \([4]\) | \(307200\) | \(2.5458\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8470.ba have rank \(1\).
Complex multiplication
The elliptic curves in class 8470.ba do not have complex multiplication.Modular form 8470.2.a.ba
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.