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SageMath
E = EllipticCurve("ho1")
E.isogeny_class()
Elliptic curves in class 369600ho
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 369600.ho5 | 369600ho1 | \([0, -1, 0, 230467, -140326563]\) | \(84611246065664/580054565475\) | \(-9280873047600000000\) | \([2]\) | \(6291456\) | \(2.3212\) | \(\Gamma_0(N)\)-optimal |
| 369600.ho4 | 369600ho2 | \([0, -1, 0, -3050033, -1869150063]\) | \(12257375872392016/1191317675625\) | \(304977324960000000000\) | \([2, 2]\) | \(12582912\) | \(2.6678\) | |
| 369600.ho3 | 369600ho3 | \([0, -1, 0, -10988033, 11950907937]\) | \(143279368983686884/22699269140625\) | \(23244051600000000000000\) | \([2, 2]\) | \(25165824\) | \(3.0144\) | |
| 369600.ho2 | 369600ho4 | \([0, -1, 0, -47600033, -126386400063]\) | \(11647843478225136004/128410942275\) | \(131492804889600000000\) | \([2]\) | \(25165824\) | \(3.0144\) | |
| 369600.ho1 | 369600ho5 | \([0, -1, 0, -168488033, 841818407937]\) | \(258286045443018193442/8440380939375\) | \(17285900163840000000000\) | \([2]\) | \(50331648\) | \(3.3609\) | |
| 369600.ho6 | 369600ho6 | \([0, -1, 0, 19503967, 66440111937]\) | \(400647648358480318/1163177490234375\) | \(-2382187500000000000000000\) | \([2]\) | \(50331648\) | \(3.3609\) |
Rank
sage: E.rank()
The elliptic curves in class 369600ho have rank \(1\).
Complex multiplication
The elliptic curves in class 369600ho do not have complex multiplication.Modular form 369600.2.a.ho
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
