Properties

Label 12696.t
Number of curves $4$
Conductor $12696$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("t1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 12696.t

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12696.t1 12696q3 \([0, 1, 0, -1557552, -748709472]\) \(1378334691074/69\) \(20919247546368\) \([2]\) \(135168\) \(2.0304\)  
12696.t2 12696q4 \([0, 1, 0, -160992, 5331360]\) \(1522096994/839523\) \(254524484896659456\) \([2]\) \(135168\) \(2.0304\)  
12696.t3 12696q2 \([0, 1, 0, -97512, -11681280]\) \(676449508/4761\) \(721714040349696\) \([2, 2]\) \(67584\) \(1.6838\)  
12696.t4 12696q1 \([0, 1, 0, -2292, -407232]\) \(-35152/1863\) \(-70602460468992\) \([4]\) \(33792\) \(1.3373\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 12696.t have rank \(0\).

Complex multiplication

The elliptic curves in class 12696.t do not have complex multiplication.

Modular form 12696.2.a.t

sage: E.q_eigenform(10)
 
\(q + q^{3} + 2q^{5} + 4q^{7} + q^{9} - 2q^{13} + 2q^{15} + 2q^{17} + 4q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.