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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 12696.t
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12696.t1 | 12696q3 | \([0, 1, 0, -1557552, -748709472]\) | \(1378334691074/69\) | \(20919247546368\) | \([2]\) | \(135168\) | \(2.0304\) | |
12696.t2 | 12696q4 | \([0, 1, 0, -160992, 5331360]\) | \(1522096994/839523\) | \(254524484896659456\) | \([2]\) | \(135168\) | \(2.0304\) | |
12696.t3 | 12696q2 | \([0, 1, 0, -97512, -11681280]\) | \(676449508/4761\) | \(721714040349696\) | \([2, 2]\) | \(67584\) | \(1.6838\) | |
12696.t4 | 12696q1 | \([0, 1, 0, -2292, -407232]\) | \(-35152/1863\) | \(-70602460468992\) | \([4]\) | \(33792\) | \(1.3373\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12696.t have rank \(0\).
Complex multiplication
The elliptic curves in class 12696.t do not have complex multiplication.Modular form 12696.2.a.t
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.