Properties

Label 53312.m
Number of curves $2$
Conductor $53312$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("m1")
 
E.isogeny_class()
 

Elliptic curves in class 53312.m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
53312.m1 53312cg2 \([0, 1, 0, -9473, 336895]\) \(6097250/289\) \(4456521531392\) \([2]\) \(98304\) \(1.1877\)  
53312.m2 53312cg1 \([0, 1, 0, -1633, -19041]\) \(62500/17\) \(131074162688\) \([2]\) \(49152\) \(0.84111\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 53312.m have rank \(2\).

Complex multiplication

The elliptic curves in class 53312.m do not have complex multiplication.

Modular form 53312.2.a.m

sage: E.q_eigenform(10)
 
\(q - 2 q^{3} + q^{9} + 2 q^{11} - 6 q^{13} + q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.