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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 12696.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12696.g1 | 12696o3 | \([0, -1, 0, -397984, -81644900]\) | \(45989074372/7555707\) | \(1145360182034967552\) | \([2]\) | \(202752\) | \(2.1868\) | |
12696.g2 | 12696o2 | \([0, -1, 0, -112324, 13308484]\) | \(4135597648/385641\) | \(14614709317081344\) | \([2, 2]\) | \(101376\) | \(1.8402\) | |
12696.g3 | 12696o1 | \([0, -1, 0, -109679, 14017344]\) | \(61604313088/621\) | \(1470884593104\) | \([4]\) | \(50688\) | \(1.4937\) | \(\Gamma_0(N)\)-optimal |
12696.g4 | 12696o4 | \([0, -1, 0, 131016, 62852508]\) | \(1640689628/12223143\) | \(-1852890972548226048\) | \([2]\) | \(202752\) | \(2.1868\) |
Rank
sage: E.rank()
The elliptic curves in class 12696.g have rank \(1\).
Complex multiplication
The elliptic curves in class 12696.g do not have complex multiplication.Modular form 12696.2.a.g
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.