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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 96600i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
96600.d6 | 96600i1 | \([0, -1, 0, 57617, 1636012]\) | \(84611246065664/53699121315\) | \(-13424780328750000\) | \([2]\) | \(688128\) | \(1.7837\) | \(\Gamma_0(N)\)-optimal |
96600.d5 | 96600i2 | \([0, -1, 0, -242508, 13641012]\) | \(394315384276816/208332909225\) | \(833331636900000000\) | \([2, 2]\) | \(1376256\) | \(2.1302\) | |
96600.d3 | 96600i3 | \([0, -1, 0, -2227008, -1268345988]\) | \(76343005935514084/694180580625\) | \(11106889290000000000\) | \([2, 2]\) | \(2752512\) | \(2.4768\) | |
96600.d2 | 96600i4 | \([0, -1, 0, -3060008, 2059146012]\) | \(198048499826486404/242568272835\) | \(3881092365360000000\) | \([2]\) | \(2752512\) | \(2.4768\) | |
96600.d4 | 96600i5 | \([0, -1, 0, -652008, -3029195988]\) | \(-957928673903042/123339801817575\) | \(-3946873658162400000000\) | \([2]\) | \(5505024\) | \(2.8234\) | |
96600.d1 | 96600i6 | \([0, -1, 0, -35554008, -81586415988]\) | \(155324313723954725282/13018359375\) | \(416587500000000000\) | \([2]\) | \(5505024\) | \(2.8234\) |
Rank
sage: E.rank()
The elliptic curves in class 96600i have rank \(0\).
Complex multiplication
The elliptic curves in class 96600i do not have complex multiplication.Modular form 96600.2.a.i
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.