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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 330e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 330.b3 | 330e1 | \([1, 1, 0, -22, -44]\) | \(1263214441/211200\) | \(211200\) | \([2]\) | \(64\) | \(-0.25715\) | \(\Gamma_0(N)\)-optimal |
| 330.b2 | 330e2 | \([1, 1, 0, -102, 324]\) | \(119168121961/10890000\) | \(10890000\) | \([2, 2]\) | \(128\) | \(0.089420\) | |
| 330.b1 | 330e3 | \([1, 1, 0, -1602, 24024]\) | \(455129268177961/4392300\) | \(4392300\) | \([4]\) | \(256\) | \(0.43599\) | |
| 330.b4 | 330e4 | \([1, 1, 0, 118, 1776]\) | \(179310732119/1392187500\) | \(-1392187500\) | \([2]\) | \(256\) | \(0.43599\) |
Rank
sage: E.rank()
The elliptic curves in class 330e have rank \(1\).
Complex multiplication
The elliptic curves in class 330e do not have complex multiplication.Modular form 330.2.a.e
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.
