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Properties

Label 630i
Number of curves $8$
Conductor $630$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("i1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 630i have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(7\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 630i do not have complex multiplication.

Modular form 630.2.a.i

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - q^{10} + 2 q^{13} + q^{14} + q^{16} + 6 q^{17} + 8 q^{19} + O(q^{20})\) 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 630i

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
630.h7 630i1 \([1, -1, 1, -4478, -114163]\) \(13619385906841/6048000\) \(4408992000\) \([2]\) \(768\) \(0.80912\) \(\Gamma_0(N)\)-optimal
630.h6 630i2 \([1, -1, 1, -5198, -74419]\) \(21302308926361/8930250000\) \(6510152250000\) \([2, 2]\) \(1536\) \(1.1557\)  
630.h5 630i3 \([1, -1, 1, -13253, 449597]\) \(353108405631241/86318776320\) \(62926387937280\) \([6]\) \(2304\) \(1.3584\)  
630.h4 630i4 \([1, -1, 1, -39218, 2946557]\) \(9150443179640281/184570312500\) \(134551757812500\) \([2]\) \(3072\) \(1.5023\)  
630.h8 630i5 \([1, -1, 1, 17302, -560419]\) \(785793873833639/637994920500\) \(-465098297044500\) \([2]\) \(3072\) \(1.5023\)  
630.h2 630i6 \([1, -1, 1, -197573, 33848381]\) \(1169975873419524361/108425318400\) \(79042057113600\) \([2, 6]\) \(4608\) \(1.7050\)  
630.h1 630i7 \([1, -1, 1, -3161093, 2164026557]\) \(4791901410190533590281/41160000\) \(30005640000\) \([6]\) \(9216\) \(2.0516\)  
630.h3 630i8 \([1, -1, 1, -183173, 38980541]\) \(-932348627918877961/358766164249920\) \(-261540533738191680\) \([6]\) \(9216\) \(2.0516\)