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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("va1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 235200.2.a.va

Copy content 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

Copy content 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

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 235200va

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
235200.va7 235200va1 \([0, 1, 0, -3216033, -782171937]\) \(7633736209/3870720\) \(1865262437498880000000\) \([2]\) \(10616832\) \(2.7740\) \(\Gamma_0(N)\)-optimal
235200.va5 235200va2 \([0, 1, 0, -28304033, 57396900063]\) \(5203798902289/57153600\) \(27541765678694400000000\) \([2, 2]\) \(21233664\) \(3.1205\)  
235200.va4 235200va3 \([0, 1, 0, -210192033, -1173000155937]\) \(2131200347946769/2058000\) \(991730245632000000000\) \([2]\) \(31850496\) \(3.3233\)  
235200.va2 235200va4 \([0, 1, 0, -451664033, 3694482660063]\) \(21145699168383889/2593080\) \(1249580109496320000000\) \([2]\) \(42467328\) \(3.4671\)  
235200.va6 235200va5 \([0, 1, 0, -6352033, 144173156063]\) \(-58818484369/18600435000\) \(-8963369276682240000000000\) \([2]\) \(42467328\) \(3.4671\)  
235200.va3 235200va6 \([0, 1, 0, -211760033, -1154612219937]\) \(2179252305146449/66177562500\) \(31890325711104000000000000\) \([2, 2]\) \(63700992\) \(3.6698\)  
235200.va1 235200va7 \([0, 1, 0, -505760033, 2754117780063]\) \(29689921233686449/10380965400750\) \(5002486572780899328000000000\) \([2]\) \(127401984\) \(4.0164\)  
235200.va8 235200va8 \([0, 1, 0, 57151967, -3886489227937]\) \(42841933504271/13565917968750\) \(-6537284334000000000000000000\) \([2]\) \(127401984\) \(4.0164\)