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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 91035.bi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
91035.bi1 | 91035h4 | \([1, -1, 0, -4385340, -3533456039]\) | \(530044731605089/26309115\) | \(462942759329606115\) | \([2]\) | \(2359296\) | \(2.4611\) | |
91035.bi2 | 91035h3 | \([1, -1, 0, -1394190, 589363051]\) | \(17032120495489/1339001685\) | \(23561459015283944685\) | \([2]\) | \(2359296\) | \(2.4611\) | |
91035.bi3 | 91035h2 | \([1, -1, 0, -288765, -48909344]\) | \(151334226289/28676025\) | \(504591588888671025\) | \([2, 2]\) | \(1179648\) | \(2.1145\) | |
91035.bi4 | 91035h1 | \([1, -1, 0, 36360, -4497269]\) | \(302111711/669375\) | \(-11778515146794375\) | \([2]\) | \(589824\) | \(1.7679\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 91035.bi have rank \(0\).
Complex multiplication
The elliptic curves in class 91035.bi do not have complex multiplication.Modular form 91035.2.a.bi
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.