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SageMath
E = EllipticCurve("gr1")
E.isogeny_class()
Elliptic curves in class 405720.gr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
405720.gr1 | 405720gr6 | \([0, 0, 0, -627172707, -6045452429506]\) | \(155324313723954725282/13018359375\) | \(2286662011653600000000\) | \([2]\) | \(88080384\) | \(3.5409\) | |
405720.gr2 | 405720gr3 | \([0, 0, 0, -53978547, 152482439774]\) | \(198048499826486404/242568272835\) | \(21303439194857085987840\) | \([2]\) | \(44040192\) | \(3.1943\) | \(\Gamma_0(N)\)-optimal* |
405720.gr3 | 405720gr4 | \([0, 0, 0, -39284427, -94024215754]\) | \(76343005935514084/694180580625\) | \(60966067889903621760000\) | \([2, 2]\) | \(44040192\) | \(3.1943\) | |
405720.gr4 | 405720gr5 | \([0, 0, 0, -11501427, -224443174354]\) | \(-957928673903042/123339801817575\) | \(-21664514799210818307225600\) | \([2]\) | \(88080384\) | \(3.5409\) | |
405720.gr5 | 405720gr2 | \([0, 0, 0, -4277847, 1004646314]\) | \(394315384276816/208332909225\) | \(4574183808223581753600\) | \([2, 2]\) | \(22020096\) | \(2.8478\) | \(\Gamma_0(N)\)-optimal* |
405720.gr6 | 405720gr1 | \([0, 0, 0, 1016358, 122631761]\) | \(84611246065664/53699121315\) | \(-73689045380735505840\) | \([2]\) | \(11010048\) | \(2.5012\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 405720.gr have rank \(1\).
Complex multiplication
The elliptic curves in class 405720.gr do not have complex multiplication.Modular form 405720.2.a.gr
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.