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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 17325z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17325.r4 | 17325z1 | \([1, -1, 1, -8255, -386378]\) | \(-5461074081/2562175\) | \(-29184774609375\) | \([2]\) | \(49152\) | \(1.2898\) | \(\Gamma_0(N)\)-optimal |
| 17325.r3 | 17325z2 | \([1, -1, 1, -144380, -21077378]\) | \(29220958012401/3705625\) | \(42209384765625\) | \([2, 2]\) | \(98304\) | \(1.6364\) | |
| 17325.r1 | 17325z3 | \([1, -1, 1, -2310005, -1350771128]\) | \(119678115308998401/1925\) | \(21926953125\) | \([2]\) | \(196608\) | \(1.9829\) | |
| 17325.r2 | 17325z4 | \([1, -1, 1, -156755, -17241128]\) | \(37397086385121/10316796875\) | \(117514764404296875\) | \([2]\) | \(196608\) | \(1.9829\) |
Rank
sage: E.rank()
The elliptic curves in class 17325z have rank \(1\).
Complex multiplication
The elliptic curves in class 17325z do not have complex multiplication.Modular form 17325.2.a.z
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.
