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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 28830h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
28830.a8 | 28830h1 | \([1, 1, 0, 1422, -62748]\) | \(357911/2160\) | \(-1917007950960\) | \([2]\) | \(57600\) | \(1.0387\) | \(\Gamma_0(N)\)-optimal |
28830.a6 | 28830h2 | \([1, 1, 0, -17798, -835392]\) | \(702595369/72900\) | \(64699018344900\) | \([2, 2]\) | \(115200\) | \(1.3853\) | |
28830.a7 | 28830h3 | \([1, 1, 0, -12993, 1860213]\) | \(-273359449/1536000\) | \(-1363205654016000\) | \([2]\) | \(172800\) | \(1.5880\) | |
28830.a5 | 28830h4 | \([1, 1, 0, -65848, 5574478]\) | \(35578826569/5314410\) | \(4716558437343210\) | \([2]\) | \(230400\) | \(1.7319\) | |
28830.a4 | 28830h5 | \([1, 1, 0, -277268, -56310078]\) | \(2656166199049/33750\) | \(29953249233750\) | \([2]\) | \(230400\) | \(1.7319\) | |
28830.a3 | 28830h6 | \([1, 1, 0, -320513, 69576117]\) | \(4102915888729/9000000\) | \(7987533129000000\) | \([2, 2]\) | \(345600\) | \(1.9346\) | |
28830.a1 | 28830h7 | \([1, 1, 0, -5125513, 4464229117]\) | \(16778985534208729/81000\) | \(71887798161000\) | \([2]\) | \(691200\) | \(2.2812\) | |
28830.a2 | 28830h8 | \([1, 1, 0, -435833, 14891373]\) | \(10316097499609/5859375000\) | \(5200216880859375000\) | \([2]\) | \(691200\) | \(2.2812\) |
Rank
sage: E.rank()
The elliptic curves in class 28830h have rank \(2\).
Complex multiplication
The elliptic curves in class 28830h do not have complex multiplication.Modular form 28830.2.a.h
sage: E.q_eigenform(10)
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.