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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 12870q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12870.b4 | 12870q1 | \([1, -1, 0, 585, 6925]\) | \(30342134159/47190000\) | \(-34401510000\) | \([2]\) | \(16384\) | \(0.70705\) | \(\Gamma_0(N)\)-optimal |
12870.b3 | 12870q2 | \([1, -1, 0, -3915, 72625]\) | \(9104453457841/2226896100\) | \(1623407256900\) | \([2, 2]\) | \(32768\) | \(1.0536\) | |
12870.b2 | 12870q3 | \([1, -1, 0, -21465, -1145345]\) | \(1500376464746641/83599963590\) | \(60944373457110\) | \([2]\) | \(65536\) | \(1.4002\) | |
12870.b1 | 12870q4 | \([1, -1, 0, -58365, 5441395]\) | \(30161840495801041/2799263610\) | \(2040663171690\) | \([2]\) | \(65536\) | \(1.4002\) |
Rank
sage: E.rank()
The elliptic curves in class 12870q have rank \(2\).
Complex multiplication
The elliptic curves in class 12870q do not have complex multiplication.Modular form 12870.2.a.q
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 Cremona 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 Cremona labels.