↑

Properties

Label 431970.gf
Number of curves $8$
Conductor $431970$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("gf1")
 
E.isogeny_class()
 

Elliptic curves in class 431970.gf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
431970.gf1 431970gf8 \([1, 0, 0, -21163276131, -961718528520639]\) \(591720065532918583239955136329/116891407012939453125000000\) \(207080257899250030517578125000000\) \([2]\) \(1911029760\) \(4.9173\)  
431970.gf2 431970gf5 \([1, 0, 0, -20035034016, -1091525533139700]\) \(502039459750388822744052370969/6444603154532812500\) \(11417007609047303845312500\) \([2]\) \(637009920\) \(4.3680\)  
431970.gf3 431970gf6 \([1, 0, 0, -6515461411, 188928838557185]\) \(17266453047612484705388895049/1288004819409000000000000\) \(2281779105877027449000000000000\) \([2, 2]\) \(955514880\) \(4.5708\)  
431970.gf4 431970gf3 \([1, 0, 0, -6394035491, 196791871147521]\) \(16318969429297971769640983369/102045248126976000000\) \(180779381817073729536000000\) \([2]\) \(477757440\) \(4.2242\) \(\Gamma_0(N)\)-optimal*
431970.gf5 431970gf2 \([1, 0, 0, -1253266396, -17024363953024]\) \(122884692280581205924284889/439106354595306090000\) \(777903692653215052106490000\) \([2, 2]\) \(318504960\) \(4.0215\)  
431970.gf6 431970gf4 \([1, 0, 0, -692975896, -32323320171724]\) \(-20774088968758822168212889/242753662862303369030100\) \(-430052921734005018742332986100\) \([2]\) \(637009920\) \(4.3680\)  
431970.gf7 431970gf1 \([1, 0, 0, -114424076, 3834184080]\) \(93523304529581769096409/54118679989886265600\) \(95874542841562902572601600\) \([2]\) \(159252480\) \(3.6749\) \(\Gamma_0(N)\)-optimal*
431970.gf8 431970gf7 \([1, 0, 0, 6189538589, 836342605557185]\) \(14802750729576629005731104951/179133615680899546821000000\) \(-317346127329270082065757581000000\) \([2]\) \(1911029760\) \(4.9173\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 431970.gf1.

Rank

sage: E.rank()
 

The elliptic curves in class 431970.gf have rank \(1\).

Complex multiplication

The elliptic curves in class 431970.gf do not have complex multiplication.

Modular form 431970.2.a.gf

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - q^{7} + q^{8} + q^{9} - q^{10} + q^{12} - 2 q^{13} - q^{14} - q^{15} + q^{16} + q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.