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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 114a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
114.c3 | 114a1 | \([1, 0, 0, -8, 0]\) | \(57066625/32832\) | \(32832\) | \([6]\) | \(12\) | \(-0.44380\) | \(\Gamma_0(N)\)-optimal |
114.c4 | 114a2 | \([1, 0, 0, 32, 8]\) | \(3616805375/2105352\) | \(-2105352\) | \([6]\) | \(24\) | \(-0.097231\) | |
114.c1 | 114a3 | \([1, 0, 0, -428, -3444]\) | \(8671983378625/82308\) | \(82308\) | \([2]\) | \(36\) | \(0.10550\) | |
114.c2 | 114a4 | \([1, 0, 0, -418, -3610]\) | \(-8078253774625/846825858\) | \(-846825858\) | \([2]\) | \(72\) | \(0.45208\) |
Rank
sage: E.rank()
The elliptic curves in class 114a have rank \(0\).
Complex multiplication
The elliptic curves in class 114a do not have complex multiplication.Modular form 114.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.