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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 143430b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
143430.u2 | 143430b1 | \([1, 0, 0, -505833255, 4487049903177]\) | \(-14313500104002032435046925025521/414610368065961984000000000\) | \(-414610368065961984000000000\) | \([9]\) | \(87340032\) | \(3.8867\) | \(\Gamma_0(N)\)-optimal |
143430.u3 | 143430b2 | \([1, 0, 0, 2284886745, 17553247599177]\) | \(1319221579871651050877529633054479/896541855128180128429337664000\) | \(-896541855128180128429337664000\) | \([3]\) | \(262020096\) | \(4.4360\) | |
143430.u1 | 143430b3 | \([1, 0, 0, -25739105055, -1892948058681663]\) | \(-1885836510340587452798601371917380721/456620935285821819761620060937640\) | \(-456620935285821819761620060937640\) | \([]\) | \(786060288\) | \(4.9853\) |
Rank
sage: E.rank()
The elliptic curves in class 143430b have rank \(1\).
Complex multiplication
The elliptic curves in class 143430b do not have complex multiplication.Modular form 143430.2.a.b
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.