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SageMath
E = EllipticCurve("hr1")
E.isogeny_class()
Elliptic curves in class 154560hr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 154560.b6 | 154560hr1 | \([0, -1, 0, -4821761, -443895039]\) | \(47293441677949844161/27041817600000000\) | \(7088850232934400000000\) | \([2]\) | \(9437184\) | \(2.8825\) | \(\Gamma_0(N)\)-optimal |
| 154560.b4 | 154560hr2 | \([0, -1, 0, -56021761, -161037815039]\) | \(74174404299602673044161/178530248806560000\) | \(46800633543146864640000\) | \([2, 2]\) | \(18874368\) | \(3.2290\) | |
| 154560.b5 | 154560hr3 | \([0, -1, 0, -35413761, -281046442239]\) | \(-18736995756767139956161/119334500162058560400\) | \(-31282823210482679257497600\) | \([2]\) | \(37748736\) | \(3.5756\) | |
| 154560.b2 | 154560hr4 | \([0, -1, 0, -895829761, -10319859267839]\) | \(303291507481995500913332161/1763329743680400\) | \(462246312327354777600\) | \([2, 2]\) | \(37748736\) | \(3.5756\) | |
| 154560.b3 | 154560hr5 | \([0, -1, 0, -895311361, -10332400089599]\) | \(-302765284673144739899429761/731344538939408411220\) | \(-191717582815732278550855680\) | \([2]\) | \(75497472\) | \(3.9222\) | |
| 154560.b1 | 154560hr6 | \([0, -1, 0, -14333276161, -660485328353279]\) | \(1242282009445982549834550082561/41992020\) | \(11007956090880\) | \([2]\) | \(75497472\) | \(3.9222\) |
Rank
sage: E.rank()
The elliptic curves in class 154560hr have rank \(1\).
Complex multiplication
The elliptic curves in class 154560hr do not have complex multiplication.Modular form 154560.2.a.hr
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.
