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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 3192d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3192.g2 | 3192d1 | \([0, -1, 0, -23948, 1131348]\) | \(5933482010818000/1304188224633\) | \(333872185506048\) | \([2]\) | \(9600\) | \(1.5003\) | \(\Gamma_0(N)\)-optimal |
3192.g1 | 3192d2 | \([0, -1, 0, -360088, 83283964]\) | \(5042558062190438500/358269592023\) | \(366868062231552\) | \([2]\) | \(19200\) | \(1.8469\) |
Rank
sage: E.rank()
The elliptic curves in class 3192d have rank \(1\).
Complex multiplication
The elliptic curves in class 3192d do not have complex multiplication.Modular form 3192.2.a.d
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.