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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 12705.l
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12705.l1 | 12705g5 | \([1, 1, 0, -1603252, 780654241]\) | \(257260669489908001/14267882475\) | \(25276424145293475\) | \([4]\) | \(245760\) | \(2.2128\) | |
12705.l2 | 12705g3 | \([1, 1, 0, -105877, 10704016]\) | \(74093292126001/14707625625\) | \(26055455959850625\) | \([2, 2]\) | \(122880\) | \(1.8662\) | |
12705.l3 | 12705g2 | \([1, 1, 0, -32672, -2136141]\) | \(2177286259681/161417025\) | \(285960106226025\) | \([2, 2]\) | \(61440\) | \(1.5197\) | |
12705.l4 | 12705g1 | \([1, 1, 0, -32067, -2223624]\) | \(2058561081361/12705\) | \(22507682505\) | \([2]\) | \(30720\) | \(1.1731\) | \(\Gamma_0(N)\)-optimal |
12705.l5 | 12705g4 | \([1, 1, 0, 30853, -9365286]\) | \(1833318007919/22507682505\) | \(-39873732526240305\) | \([2]\) | \(122880\) | \(1.8662\) | |
12705.l6 | 12705g6 | \([1, 1, 0, 220218, 63987939]\) | \(666688497209279/1381398046875\) | \(-2447230905319921875\) | \([2]\) | \(245760\) | \(2.2128\) |
Rank
sage: E.rank()
The elliptic curves in class 12705.l have rank \(0\).
Complex multiplication
The elliptic curves in class 12705.l do not have complex multiplication.Modular form 12705.2.a.l
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.