Properties

Label 235200gk
Number of curves $8$
Conductor $235200$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("gk1")
 
E.isogeny_class()
 

Elliptic curves in class 235200gk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
235200.gk7 235200gk1 \([0, -1, 0, -3216033, 782171937]\) \(7633736209/3870720\) \(1865262437498880000000\) \([2]\) \(10616832\) \(2.7740\) \(\Gamma_0(N)\)-optimal
235200.gk5 235200gk2 \([0, -1, 0, -28304033, -57396900063]\) \(5203798902289/57153600\) \(27541765678694400000000\) \([2, 2]\) \(21233664\) \(3.1205\)  
235200.gk4 235200gk3 \([0, -1, 0, -210192033, 1173000155937]\) \(2131200347946769/2058000\) \(991730245632000000000\) \([2]\) \(31850496\) \(3.3233\)  
235200.gk6 235200gk4 \([0, -1, 0, -6352033, -144173156063]\) \(-58818484369/18600435000\) \(-8963369276682240000000000\) \([2]\) \(42467328\) \(3.4671\)  
235200.gk2 235200gk5 \([0, -1, 0, -451664033, -3694482660063]\) \(21145699168383889/2593080\) \(1249580109496320000000\) \([2]\) \(42467328\) \(3.4671\)  
235200.gk3 235200gk6 \([0, -1, 0, -211760033, 1154612219937]\) \(2179252305146449/66177562500\) \(31890325711104000000000000\) \([2, 2]\) \(63700992\) \(3.6698\)  
235200.gk8 235200gk7 \([0, -1, 0, 57151967, 3886489227937]\) \(42841933504271/13565917968750\) \(-6537284334000000000000000000\) \([2]\) \(127401984\) \(4.0164\)  
235200.gk1 235200gk8 \([0, -1, 0, -505760033, -2754117780063]\) \(29689921233686449/10380965400750\) \(5002486572780899328000000000\) \([2]\) \(127401984\) \(4.0164\)  

Rank

sage: E.rank()
 

The elliptic curves in class 235200gk have rank \(0\).

Complex multiplication

The elliptic curves in class 235200gk do not have complex multiplication.

Modular form 235200.2.a.gk

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

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.