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SageMath
E = EllipticCurve("ip1")
E.isogeny_class()
Elliptic curves in class 277200.ip
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
277200.ip1 | 277200ip4 | \([0, 0, 0, -12672075, -17268047750]\) | \(4823468134087681/30382271150\) | \(1417515242774400000000\) | \([2]\) | \(15925248\) | \(2.8960\) | |
277200.ip2 | 277200ip2 | \([0, 0, 0, -972075, 353052250]\) | \(2177286259681/105875000\) | \(4939704000000000000\) | \([2]\) | \(5308416\) | \(2.3467\) | |
277200.ip3 | 277200ip3 | \([0, 0, 0, -324075, -585899750]\) | \(-80677568161/3131816380\) | \(-146118025025280000000\) | \([2]\) | \(7962624\) | \(2.5494\) | |
277200.ip4 | 277200ip1 | \([0, 0, 0, 35925, 21420250]\) | \(109902239/4312000\) | \(-201180672000000000\) | \([2]\) | \(2654208\) | \(2.0001\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 277200.ip have rank \(0\).
Complex multiplication
The elliptic curves in class 277200.ip do not have complex multiplication.Modular form 277200.2.a.ip
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.