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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 630f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
630.f7 | 630f1 | \([1, -1, 0, -369, 1053]\) | \(7633736209/3870720\) | \(2821754880\) | \([2]\) | \(384\) | \(0.50588\) | \(\Gamma_0(N)\)-optimal |
630.f5 | 630f2 | \([1, -1, 0, -3249, -69795]\) | \(5203798902289/57153600\) | \(41664974400\) | \([2, 2]\) | \(768\) | \(0.85245\) | |
630.f4 | 630f3 | \([1, -1, 0, -24129, 1448685]\) | \(2131200347946769/2058000\) | \(1500282000\) | \([6]\) | \(1152\) | \(1.0552\) | |
630.f2 | 630f4 | \([1, -1, 0, -51849, -4531275]\) | \(21145699168383889/2593080\) | \(1890355320\) | \([2]\) | \(1536\) | \(1.1990\) | |
630.f6 | 630f5 | \([1, -1, 0, -729, -177147]\) | \(-58818484369/18600435000\) | \(-13559717115000\) | \([2]\) | \(1536\) | \(1.1990\) | |
630.f3 | 630f6 | \([1, -1, 0, -24309, 1426113]\) | \(2179252305146449/66177562500\) | \(48243443062500\) | \([2, 6]\) | \(2304\) | \(1.4018\) | |
630.f1 | 630f7 | \([1, -1, 0, -58059, -3373137]\) | \(29689921233686449/10380965400750\) | \(7567723777146750\) | \([6]\) | \(4608\) | \(1.7483\) | |
630.f8 | 630f8 | \([1, -1, 0, 6561, 4778595]\) | \(42841933504271/13565917968750\) | \(-9889554199218750\) | \([6]\) | \(4608\) | \(1.7483\) |
Rank
sage: E.rank()
The elliptic curves in class 630f have rank \(0\).
Complex multiplication
The elliptic curves in class 630f do not have complex multiplication.Modular form 630.2.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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.