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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 450840g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
450840.g4 | 450840g1 | \([0, -1, 0, -4790271, 4037012496]\) | \(31476797652269056/49725\) | \(19203849896400\) | \([2]\) | \(6488064\) | \(2.2433\) | \(\Gamma_0(N)\)-optimal |
450840.g3 | 450840g2 | \([0, -1, 0, -4791716, 4034456580]\) | \(1969080716416336/2472575625\) | \(15278582977575840000\) | \([2, 2]\) | \(12976128\) | \(2.5898\) | |
450840.g5 | 450840g3 | \([0, -1, 0, -3514336, 6233083036]\) | \(-194204905090564/566398828125\) | \(-13999606574475600000000\) | \([2]\) | \(25952256\) | \(2.9364\) | |
450840.g2 | 450840g4 | \([0, -1, 0, -6092216, 1672228380]\) | \(1011710313226084/536724738225\) | \(13266155932582784025600\) | \([2, 2]\) | \(25952256\) | \(2.9364\) | |
450840.g6 | 450840g5 | \([0, -1, 0, 23212384, 13054135020]\) | \(27980756504588158/17683545112935\) | \(-874163794591910615070720\) | \([2]\) | \(51904512\) | \(3.2830\) | |
450840.g1 | 450840g6 | \([0, -1, 0, -56204816, -160913091060]\) | \(397210600760070242/3536192675535\) | \(174807233951786341201920\) | \([2]\) | \(51904512\) | \(3.2830\) |
Rank
sage: E.rank()
The elliptic curves in class 450840g have rank \(1\).
Complex multiplication
The elliptic curves in class 450840g do not have complex multiplication.Modular form 450840.2.a.g
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.