Properties

Label 49.a
Number of curves $4$
Conductor $49$
CM \(\Q(\sqrt{-7}) \)
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 49.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
49.a1 49a4 \([1, -1, 0, -1822, 30393]\) \(16581375\) \(40353607\) \([2]\) \(14\) \(0.52039\)   \(-28\)
49.a2 49a3 \([1, -1, 0, -107, 552]\) \(-3375\) \(-40353607\) \([2]\) \(7\) \(0.17382\)   \(-7\)
49.a3 49a2 \([1, -1, 0, -37, -78]\) \(16581375\) \(343\) \([2]\) \(2\) \(-0.45256\)   \(-28\)
49.a4 49a1 \([1, -1, 0, -2, -1]\) \(-3375\) \(-343\) \([2]\) \(1\) \(-0.79914\) \(\Gamma_0(N)\)-optimal \(-7\)

Rank

sage: E.rank()
 

The elliptic curves in class 49.a have rank \(0\).

Complex multiplication

Each elliptic curve in class 49.a 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)
 
\(q + q^{2} - q^{4} - 3 q^{8} - 3 q^{9} + 4 q^{11} - q^{16} - 3 q^{18} + O(q^{20})\) Copy content Toggle raw display

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.