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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 7728.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
7728.d1 | 7728i1 | \([0, -1, 0, -478493, -127241151]\) | \(-47327266415721472000/1222082060283\) | \(-312853007432448\) | \([]\) | \(47520\) | \(1.8882\) | \(\Gamma_0(N)\)-optimal |
7728.d2 | 7728i2 | \([0, -1, 0, -148013, -299020767]\) | \(-1400832679220224000/150124273180279587\) | \(-38431813934151574272\) | \([]\) | \(142560\) | \(2.4375\) |
Rank
sage: E.rank()
The elliptic curves in class 7728.d have rank \(1\).
Complex multiplication
The elliptic curves in class 7728.d do not have complex multiplication.Modular form 7728.2.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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.