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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 30912p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
30912.h2 | 30912p1 | \([0, -1, 0, -28329, -2765655]\) | \(-613864936718272/456986789343\) | \(-1871817889148928\) | \([2]\) | \(156672\) | \(1.6298\) | \(\Gamma_0(N)\)-optimal |
30912.h1 | 30912p2 | \([0, -1, 0, -515009, -142053471]\) | \(461019267341732744/115946266023\) | \(3799327245041664\) | \([2]\) | \(313344\) | \(1.9763\) |
Rank
sage: E.rank()
The elliptic curves in class 30912p have rank \(1\).
Complex multiplication
The elliptic curves in class 30912p do not have complex multiplication.Modular form 30912.2.a.p
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.