Properties

Label 142912p
Number of curves $4$
Conductor $142912$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("p1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 142912p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
142912.bm4 142912p1 \([0, 0, 0, 37396, -3172528]\) \(22062729659823/29354283343\) \(-7695049252667392\) \([2]\) \(688128\) \(1.7341\) \(\Gamma_0(N)\)-optimal
142912.bm3 142912p2 \([0, 0, 0, -231724, -31053360]\) \(5249244962308257/1448621666569\) \(379747478161063936\) \([2, 2]\) \(1376256\) \(2.0807\)  
142912.bm2 142912p3 \([0, 0, 0, -1354604, 582039120]\) \(1048626554636928177/48569076788309\) \(12732092065594474496\) \([2]\) \(2752512\) \(2.4272\)  
142912.bm1 142912p4 \([0, 0, 0, -3414764, -2428519088]\) \(16798320881842096017/2132227789307\) \(558950721600094208\) \([2]\) \(2752512\) \(2.4272\)  

Rank

sage: E.rank()
 

The elliptic curves in class 142912p have rank \(1\).

Complex multiplication

The elliptic curves in class 142912p do not have complex multiplication.

Modular form 142912.2.a.p

sage: E.q_eigenform(10)
 
\(q + 2q^{5} + q^{7} - 3q^{9} - q^{11} - 6q^{13} - 2q^{17} - 8q^{19} + O(q^{20})\)  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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.