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SageMath
E = EllipticCurve("gv1")
E.isogeny_class()
Elliptic curves in class 106722gv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
106722.gs4 | 106722gv1 | \([1, -1, 1, -5177129, -16578743079]\) | \(-100999381393/723148272\) | \(-109875087041458428475632\) | \([2]\) | \(8847360\) | \(3.1043\) | \(\Gamma_0(N)\)-optimal |
106722.gs3 | 106722gv2 | \([1, -1, 1, -134310749, -597731686527]\) | \(1763535241378513/4612311396\) | \(700794201853268943603876\) | \([2, 2]\) | \(17694720\) | \(3.4509\) | |
106722.gs2 | 106722gv3 | \([1, -1, 1, -187138139, -83763443739]\) | \(4770223741048753/2740574865798\) | \(416402712393107262353800038\) | \([2]\) | \(35389440\) | \(3.7974\) | |
106722.gs1 | 106722gv4 | \([1, -1, 1, -2147621279, -38307037913427]\) | \(7209828390823479793/49509306\) | \(7522439749551389402586\) | \([2]\) | \(35389440\) | \(3.7974\) |
Rank
sage: E.rank()
The elliptic curves in class 106722gv have rank \(1\).
Complex multiplication
The elliptic curves in class 106722gv do not have complex multiplication.Modular form 106722.2.a.gv
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.