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SageMath
E = EllipticCurve("eq1")
E.isogeny_class()
Elliptic curves in class 250800.eq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
250800.eq1 | 250800eq8 | \([0, 1, 0, -8569713008, -303287420436012]\) | \(1087533321226184807035053481/8484255812957933638080\) | \(542992372029307752837120000000\) | \([2]\) | \(445906944\) | \(4.5341\) | |
250800.eq2 | 250800eq5 | \([0, 1, 0, -8553603008, -304491949656012]\) | \(1081411559614045490773061881/522522049500\) | \(33441411168000000000\) | \([2]\) | \(148635648\) | \(3.9848\) | |
250800.eq3 | 250800eq6 | \([0, 1, 0, -903153008, 2592990443988]\) | \(1272998045160051207059881/691293848290254950400\) | \(44242806290576316825600000000\) | \([2, 2]\) | \(222953472\) | \(4.1875\) | |
250800.eq4 | 250800eq3 | \([0, 1, 0, -698353008, 7094084843988]\) | \(588530213343917460371881/861551575695360000\) | \(55139300844503040000000000\) | \([2]\) | \(111476736\) | \(3.8410\) | |
250800.eq5 | 250800eq2 | \([0, 1, 0, -534603008, -4757767656012]\) | \(264020672568758737421881/5803468580250000\) | \(371421989136000000000000\) | \([2, 2]\) | \(74317824\) | \(3.6382\) | |
250800.eq6 | 250800eq4 | \([0, 1, 0, -515603008, -5111585656012]\) | \(-236859095231405581781881/39282983014374049500\) | \(-2514110912919939168000000000\) | \([4]\) | \(148635648\) | \(3.9848\) | |
250800.eq7 | 250800eq1 | \([0, 1, 0, -34603008, -68767656012]\) | \(71595431380957421881/9522562500000000\) | \(609444000000000000000000\) | \([2]\) | \(37158912\) | \(3.2917\) | \(\Gamma_0(N)\)-optimal |
250800.eq8 | 250800eq7 | \([0, 1, 0, 3486606992, 20406636523988]\) | \(73240740785321709623685719/45195275784938365817280\) | \(-2892497650236055412305920000000\) | \([4]\) | \(445906944\) | \(4.5341\) |
Rank
sage: E.rank()
The elliptic curves in class 250800.eq have rank \(0\).
Complex multiplication
The elliptic curves in class 250800.eq do not have complex multiplication.Modular form 250800.2.a.eq
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.