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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 49a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
49.a4 | 49a1 | \([1, -1, 0, -2, -1]\) | \(-3375\) | \(-343\) | \([2]\) | \(1\) | \(-0.79914\) | \(\Gamma_0(N)\)-optimal | \(-7\) |
49.a3 | 49a2 | \([1, -1, 0, -37, -78]\) | \(16581375\) | \(343\) | \([2]\) | \(2\) | \(-0.45256\) | \(-28\) | |
49.a2 | 49a3 | \([1, -1, 0, -107, 552]\) | \(-3375\) | \(-40353607\) | \([2]\) | \(7\) | \(0.17382\) | \(-7\) | |
49.a1 | 49a4 | \([1, -1, 0, -1822, 30393]\) | \(16581375\) | \(40353607\) | \([2]\) | \(14\) | \(0.52039\) | \(-28\) |
Rank
sage: E.rank()
The elliptic curves in class 49a have rank \(0\).
Complex multiplication
Each elliptic curve in class 49a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-7}) \).Modular form 49.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.