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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 8085.g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8085.g1 | 8085p5 | \([1, 0, 0, -649251, 201293406]\) | \(257260669489908001/14267882475\) | \(1678602105301275\) | \([2]\) | \(98304\) | \(1.9868\) | |
| 8085.g2 | 8085p3 | \([1, 0, 0, -42876, 2766231]\) | \(74093292126001/14707625625\) | \(1730337447155625\) | \([2, 2]\) | \(49152\) | \(1.6402\) | |
| 8085.g3 | 8085p2 | \([1, 0, 0, -13231, -548080]\) | \(2177286259681/161417025\) | \(18990551574225\) | \([2, 2]\) | \(24576\) | \(1.2937\) | |
| 8085.g4 | 8085p1 | \([1, 0, 0, -12986, -570669]\) | \(2058561081361/12705\) | \(1494730545\) | \([2]\) | \(12288\) | \(0.94710\) | \(\Gamma_0(N)\)-optimal |
| 8085.g5 | 8085p4 | \([1, 0, 0, 12494, -2415715]\) | \(1833318007919/22507682505\) | \(-2648006339030745\) | \([2]\) | \(49152\) | \(1.6402\) | |
| 8085.g6 | 8085p6 | \([1, 0, 0, 89179, 16473540]\) | \(666688497209279/1381398046875\) | \(-162520098816796875\) | \([2]\) | \(98304\) | \(1.9868\) |
Rank
sage: E.rank()
The elliptic curves in class 8085.g have rank \(0\).
Complex multiplication
The elliptic curves in class 8085.g do not have complex multiplication.Modular form 8085.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
