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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 3520p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3520.m3 | 3520p1 | \([0, 0, 0, -20168, -1102408]\) | \(885956203616256/15125\) | \(15488000\) | \([2]\) | \(4608\) | \(0.92153\) | \(\Gamma_0(N)\)-optimal |
| 3520.m2 | 3520p2 | \([0, 0, 0, -20188, -1100112]\) | \(55537159171536/228765625\) | \(3748096000000\) | \([2, 2]\) | \(9216\) | \(1.2681\) | |
| 3520.m1 | 3520p3 | \([0, 0, 0, -30188, 103888]\) | \(46424454082884/26794860125\) | \(1756027953152000\) | \([2]\) | \(18432\) | \(1.6147\) | |
| 3520.m4 | 3520p4 | \([0, 0, 0, -10508, -2157168]\) | \(-1957960715364/29541015625\) | \(-1936000000000000\) | \([2]\) | \(18432\) | \(1.6147\) |
Rank
sage: E.rank()
The elliptic curves in class 3520p have rank \(0\).
Complex multiplication
The elliptic curves in class 3520p do not have complex multiplication.Modular form 3520.2.a.p
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.
