Properties

Label 323400.r
Number of curves $6$
Conductor $323400$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 323400.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
323400.r1 323400r4 \([0, -1, 0, -64680408, -200198341188]\) \(15897679904620804/2475\) \(4658900400000000\) \([2]\) \(18874368\) \(2.8531\)  
323400.r2 323400r5 \([0, -1, 0, -34300408, 75850508812]\) \(1185450336504002/26043266205\) \(98046855224065440000000\) \([2]\) \(37748736\) \(3.1996\)  
323400.r3 323400r3 \([0, -1, 0, -4655408, -2115841188]\) \(5927735656804/2401490025\) \(4520526399219600000000\) \([2, 2]\) \(18874368\) \(2.8531\)  
323400.r4 323400r2 \([0, -1, 0, -4042908, -3126466188]\) \(15529488955216/6125625\) \(2882694622500000000\) \([2, 2]\) \(9437184\) \(2.5065\)  
323400.r5 323400r1 \([0, -1, 0, -214783, -63966188]\) \(-37256083456/38671875\) \(-1137426855468750000\) \([2]\) \(4718592\) \(2.1599\) \(\Gamma_0(N)\)-optimal
323400.r6 323400r6 \([0, -1, 0, 15189592, -15411991188]\) \(102949393183198/86815346805\) \(-326839639560366240000000\) \([2]\) \(37748736\) \(3.1996\)  

Rank

sage: E.rank()
 

The elliptic curves in class 323400.r have rank \(0\).

Complex multiplication

The elliptic curves in class 323400.r do not have complex multiplication.

Modular form 323400.2.a.r

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

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.