Properties

Label 30912e
Number of curves $4$
Conductor $30912$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 30912e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
30912.v4 30912e1 \([0, -1, 0, -38337, -4700223]\) \(-23771111713777/22848457968\) \(-5989586165563392\) \([2]\) \(184320\) \(1.7236\) \(\Gamma_0(N)\)-optimal
30912.v3 30912e2 \([0, -1, 0, -715457, -232618815]\) \(154502321244119857/55101928644\) \(14444639982452736\) \([2, 2]\) \(368640\) \(2.0701\)  
30912.v2 30912e3 \([0, -1, 0, -818497, -161129663]\) \(231331938231569617/90942310746882\) \(23839981108430635008\) \([2]\) \(737280\) \(2.4167\)  
30912.v1 30912e4 \([0, -1, 0, -11446337, -14901731775]\) \(632678989847546725777/80515134\) \(21106559287296\) \([2]\) \(737280\) \(2.4167\)  

Rank

sage: E.rank()
 

The elliptic curves in class 30912e have rank \(0\).

Complex multiplication

The elliptic curves in class 30912e do not have complex multiplication.

Modular form 30912.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{5} - q^{7} + q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{15} + 6 q^{17} + O(q^{20})\) 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 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.