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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 12100f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12100.j3 | 12100f1 | \([0, -1, 0, -4033, -66438]\) | \(16384/5\) | \(2214451250000\) | \([2]\) | \(17280\) | \(1.0737\) | \(\Gamma_0(N)\)-optimal |
12100.j4 | 12100f2 | \([0, -1, 0, 11092, -459688]\) | \(21296/25\) | \(-177156100000000\) | \([2]\) | \(34560\) | \(1.4203\) | |
12100.j1 | 12100f3 | \([0, -1, 0, -125033, 17055062]\) | \(488095744/125\) | \(55361281250000\) | \([2]\) | \(51840\) | \(1.6230\) | |
12100.j2 | 12100f4 | \([0, -1, 0, -109908, 21320312]\) | \(-20720464/15625\) | \(-110722562500000000\) | \([2]\) | \(103680\) | \(1.9696\) |
Rank
sage: E.rank()
The elliptic curves in class 12100f have rank \(1\).
Complex multiplication
The elliptic curves in class 12100f do not have complex multiplication.Modular form 12100.2.a.f
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.