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SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 388416.v
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
388416.v1 | 388416v5 | \([0, -1, 0, -14501249, 21259595553]\) | \(53297461115137/147\) | \(930145276526592\) | \([2]\) | \(9437184\) | \(2.5305\) | |
388416.v2 | 388416v3 | \([0, -1, 0, -906689, 332129889]\) | \(13027640977/21609\) | \(136731355649409024\) | \([2, 2]\) | \(4718592\) | \(2.1840\) | |
388416.v3 | 388416v4 | \([0, -1, 0, -721729, -234328607]\) | \(6570725617/45927\) | \(290603959966236672\) | \([2]\) | \(4718592\) | \(2.1840\) | |
388416.v4 | 388416v6 | \([0, -1, 0, -629249, 538933665]\) | \(-4354703137/17294403\) | \(-109430661638077022208\) | \([2]\) | \(9437184\) | \(2.5305\) | |
388416.v5 | 388416v2 | \([0, -1, 0, -74369, 1698849]\) | \(7189057/3969\) | \(25113922466217984\) | \([2, 2]\) | \(2359296\) | \(1.8374\) | |
388416.v6 | 388416v1 | \([0, -1, 0, 18111, 200673]\) | \(103823/63\) | \(-398633689939968\) | \([2]\) | \(1179648\) | \(1.4908\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 388416.v have rank \(2\).
Complex multiplication
The elliptic curves in class 388416.v do not have complex multiplication.Modular form 388416.2.a.v
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.