Show commands:
SageMath
E = EllipticCurve("ew1")
E.isogeny_class()
Elliptic curves in class 117600ew
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
117600.y1 | 117600ew1 | \([0, -1, 0, -257658, 41677812]\) | \(16079333824/2953125\) | \(347432203125000000\) | \([2]\) | \(1327104\) | \(2.0843\) | \(\Gamma_0(N)\)-optimal |
117600.y2 | 117600ew2 | \([0, -1, 0, 507967, 241505937]\) | \(1925134784/4465125\) | \(-33620319432000000000\) | \([2]\) | \(2654208\) | \(2.4308\) |
Rank
sage: E.rank()
The elliptic curves in class 117600ew have rank \(1\).
Complex multiplication
The elliptic curves in class 117600ew do not have complex multiplication.Modular form 117600.2.a.ew
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.