Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 3120a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3120.c3 | 3120a1 | \([0, -1, 0, -4876, -129440]\) | \(50091484483024/14625\) | \(3744000\) | \([2]\) | \(1536\) | \(0.62742\) | \(\Gamma_0(N)\)-optimal |
3120.c2 | 3120a2 | \([0, -1, 0, -4896, -128304]\) | \(12677589459076/213890625\) | \(219024000000\) | \([2, 2]\) | \(3072\) | \(0.97399\) | |
3120.c1 | 3120a3 | \([0, -1, 0, -9896, 183696]\) | \(52337949619538/23423590125\) | \(47971512576000\) | \([2]\) | \(6144\) | \(1.3206\) | |
3120.c4 | 3120a4 | \([0, -1, 0, -216, -367920]\) | \(-546718898/28564453125\) | \(-58500000000000\) | \([2]\) | \(6144\) | \(1.3206\) |
Rank
sage: E.rank()
The elliptic curves in class 3120a have rank \(1\).
Complex multiplication
The elliptic curves in class 3120a do not have complex multiplication.Modular form 3120.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.