Properties

Label 30030bt
Number of curves 88
Conductor 3003030030
CM no
Rank 11
Graph

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Show commands: SageMath
E = EllipticCurve("bt1")
 
E.isogeny_class()
 

Elliptic curves in class 30030bt

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
30030.bt7 30030bt1 [1,0,0,749461,263897441][1, 0, 0, -749461, 263897441] 46555485820017544148689/3157693080314572800-46555485820017544148689/3157693080314572800 3157693080314572800-3157693080314572800 [12][12] 774144774144 2.30122.3012 Γ0(N)\Gamma_0(N)-optimal
30030.bt6 30030bt2 [1,0,0,12180181,16360637345][1, 0, 0, -12180181, 16360637345] 199841159336796255944706769/834505270358760000199841159336796255944706769/834505270358760000 834505270358760000834505270358760000 [2,6][2, 6] 15482881548288 2.64782.6478  
30030.bt8 30030bt3 [1,0,0,4220699,295790705][1, 0, 0, 4220699, 295790705] 8315279469612171276463151/48497897968877857500008315279469612171276463151/4849789796887785750000 4849789796887785750000-4849789796887785750000 [4][4] 23224322322432 2.85052.8505  
30030.bt5 30030bt4 [1,0,0,12369181,15826636745][1, 0, 0, -12369181, 15826636745] 209289070072300727183442769/12893854589717635333800209289070072300727183442769/12893854589717635333800 1289385458971763533380012893854589717635333800 [6][6] 30965763096576 2.99442.9944  
30030.bt2 30030bt5 [1,0,0,194882701,1047131714681][1, 0, 0, -194882701, 1047131714681] 818546927584539194367471866449/14273634375000818546927584539194367471866449/14273634375000 1427363437500014273634375000 [6][6] 30965763096576 2.99442.9944  
30030.bt4 30030bt6 [1,0,0,16956121,2366883701][1, 0, 0, -16956121, 2366883701] 539142086340577084766074129/309580507925165039062500539142086340577084766074129/309580507925165039062500 309580507925165039062500309580507925165039062500 [2,2][2, 2] 46448644644864 3.19713.1971  
30030.bt3 30030bt7 [1,0,0,177737371,908394585049][1, 0, 0, -177737371, -908394585049] 620954771108295351491118574129/2882378618771462717156250620954771108295351491118574129/2882378618771462717156250 28823786187714627171562502882378618771462717156250 [2][2] 92897289289728 3.54373.5437  
30030.bt1 30030bt8 [1,0,0,195003991,1045763011475][1, 0, 0, -195003991, 1045763011475] 820076206880893214178646273009/2122496008872985839843750820076206880893214178646273009/2122496008872985839843750 21224960088729858398437502122496008872985839843750 [2][2] 92897289289728 3.54373.5437  

Rank

sage: E.rank()
 

The elliptic curves in class 30030bt have rank 11.

Complex multiplication

The elliptic curves in class 30030bt do not have complex multiplication.

Modular form 30030.2.a.bt

sage: E.q_eigenform(10)
 
q+q2+q3+q4q5+q6+q7+q8+q9q10q11+q12+q13+q14q15+q166q17+q184q19+O(q20)q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} - q^{11} + q^{12} + q^{13} + q^{14} - q^{15} + q^{16} - 6 q^{17} + q^{18} - 4 q^{19} + O(q^{20}) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The i,ji,j entry is the smallest degree of a cyclic isogeny between the ii-th and jj-th curve in the isogeny class, in the Cremona numbering.

(1234461212216223663611212244421214631242124161236326612212643122141264123241)\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.