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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 344850.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
344850.f1 | 344850f8 | \([1, 1, 0, -64808454625, -6307398167652875]\) | \(1087533321226184807035053481/8484255812957933638080\) | \(234849636129055779278291295000000\) | \([2]\) | \(2229534720\) | \(5.0399\) | |
344850.f2 | 344850f5 | \([1, 1, 0, -64686622750, -6332448672191000]\) | \(1081411559614045490773061881/522522049500\) | \(14463745070847960937500\) | \([2]\) | \(743178240\) | \(4.4906\) | |
344850.f3 | 344850f6 | \([1, 1, 0, -6830094625, 53929513187125]\) | \(1272998045160051207059881/691293848290254950400\) | \(19135456580795817971649600000000\) | \([2, 2]\) | \(1114767360\) | \(4.6933\) | |
344850.f4 | 344850f3 | \([1, 1, 0, -5281294625, 147537436387125]\) | \(588530213343917460371881/861551575695360000\) | \(23848299546725744640000000000\) | \([2]\) | \(557383680\) | \(4.3468\) | |
344850.f5 | 344850f2 | \([1, 1, 0, -4042935250, -98944677753500]\) | \(264020672568758737421881/5803468580250000\) | \(160643728148379222656250000\) | \([2, 2]\) | \(371589120\) | \(4.1440\) | |
344850.f6 | 344850f4 | \([1, 1, 0, -3899247750, -106303058316000]\) | \(-236859095231405581781881/39282983014374049500\) | \(-1087378135498867273535460937500\) | \([2]\) | \(743178240\) | \(4.4906\) | |
344850.f7 | 344850f1 | \([1, 1, 0, -261685250, -1430021503500]\) | \(71595431380957421881/9522562500000000\) | \(263590630391601562500000000\) | \([2]\) | \(185794560\) | \(3.7975\) | \(\Gamma_0(N)\)-optimal |
344850.f8 | 344850f7 | \([1, 1, 0, 26367465375, 424381085227125]\) | \(73240740785321709623685719/45195275784938365817280\) | \(-1251034186950643691962912095000000\) | \([2]\) | \(2229534720\) | \(5.0399\) |
Rank
sage: E.rank()
The elliptic curves in class 344850.f have rank \(0\).
Complex multiplication
The elliptic curves in class 344850.f do not have complex multiplication.Modular form 344850.2.a.f
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.