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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 3696q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3696.m4 | 3696q1 | \([0, -1, 0, -1552, -85568]\) | \(-100999381393/723148272\) | \(-2962015322112\) | \([2]\) | \(4608\) | \(1.0762\) | \(\Gamma_0(N)\)-optimal |
3696.m3 | 3696q2 | \([0, -1, 0, -40272, -3090240]\) | \(1763535241378513/4612311396\) | \(18892027478016\) | \([2, 2]\) | \(9216\) | \(1.4228\) | |
3696.m1 | 3696q3 | \([0, -1, 0, -643952, -198682560]\) | \(7209828390823479793/49509306\) | \(202790117376\) | \([2]\) | \(18432\) | \(1.7694\) | |
3696.m2 | 3696q4 | \([0, -1, 0, -56112, -416448]\) | \(4770223741048753/2740574865798\) | \(11225394650308608\) | \([4]\) | \(18432\) | \(1.7694\) |
Rank
sage: E.rank()
The elliptic curves in class 3696q have rank \(1\).
Complex multiplication
The elliptic curves in class 3696q do not have complex multiplication.Modular form 3696.2.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.