Show commands:
SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 135240h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
135240.dp3 | 135240h1 | \([0, 1, 0, -5700, 163680]\) | \(680136784/345\) | \(10390759680\) | \([2]\) | \(147456\) | \(0.87404\) | \(\Gamma_0(N)\)-optimal |
135240.dp2 | 135240h2 | \([0, 1, 0, -6680, 102528]\) | \(273671716/119025\) | \(14339248358400\) | \([2, 2]\) | \(294912\) | \(1.2206\) | |
135240.dp4 | 135240h3 | \([0, 1, 0, 22720, 784608]\) | \(5382838942/4197615\) | \(-1011394984212480\) | \([2]\) | \(589824\) | \(1.5672\) | |
135240.dp1 | 135240h4 | \([0, 1, 0, -51760, -4477600]\) | \(63649751618/1164375\) | \(280550511360000\) | \([2]\) | \(589824\) | \(1.5672\) |
Rank
sage: E.rank()
The elliptic curves in class 135240h have rank \(1\).
Complex multiplication
The elliptic curves in class 135240h do not have complex multiplication.Modular form 135240.2.a.h
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.