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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 180a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 180.a3 | 180a1 | \([0, 0, 0, -12, -11]\) | \(16384/5\) | \(58320\) | \([2]\) | \(12\) | \(-0.38064\) | \(\Gamma_0(N)\)-optimal |
| 180.a4 | 180a2 | \([0, 0, 0, 33, -74]\) | \(21296/25\) | \(-4665600\) | \([2]\) | \(24\) | \(-0.034070\) | |
| 180.a1 | 180a3 | \([0, 0, 0, -372, 2761]\) | \(488095744/125\) | \(1458000\) | \([6]\) | \(36\) | \(0.16866\) | |
| 180.a2 | 180a4 | \([0, 0, 0, -327, 3454]\) | \(-20720464/15625\) | \(-2916000000\) | \([6]\) | \(72\) | \(0.51524\) |
Rank
sage: E.rank()
The elliptic curves in class 180a have rank \(0\).
Complex multiplication
The elliptic curves in class 180a do not have complex multiplication.Modular form 180.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
