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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1470.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1470.b1 | 1470b7 | \([1, 1, 0, -17210393, 27473941797]\) | \(4791901410190533590281/41160000\) | \(4842432840000\) | \([2]\) | \(55296\) | \(2.4752\) | |
1470.b2 | 1470b6 | \([1, 1, 0, -1075673, 428924133]\) | \(1169975873419524361/108425318400\) | \(12756130284441600\) | \([2, 2]\) | \(27648\) | \(2.1286\) | |
1470.b3 | 1470b8 | \([1, 1, 0, -997273, 494199973]\) | \(-932348627918877961/358766164249920\) | \(-42208480457838838080\) | \([2]\) | \(55296\) | \(2.4752\) | |
1470.b4 | 1470b4 | \([1, 1, 0, -213518, 37218672]\) | \(9150443179640281/184570312500\) | \(21714512695312500\) | \([2]\) | \(18432\) | \(1.9259\) | |
1470.b5 | 1470b3 | \([1, 1, 0, -72153, 5639397]\) | \(353108405631241/86318776320\) | \(10155317715271680\) | \([2]\) | \(13824\) | \(1.7821\) | |
1470.b6 | 1470b2 | \([1, 1, 0, -28298, -973692]\) | \(21302308926361/8930250000\) | \(1050634982250000\) | \([2, 2]\) | \(9216\) | \(1.5793\) | |
1470.b7 | 1470b1 | \([1, 1, 0, -24378, -1474668]\) | \(13619385906841/6048000\) | \(711541152000\) | \([2]\) | \(4608\) | \(1.2328\) | \(\Gamma_0(N)\)-optimal |
1470.b8 | 1470b5 | \([1, 1, 0, 94202, -7025192]\) | \(785793873833639/637994920500\) | \(-75059464401904500\) | \([2]\) | \(18432\) | \(1.9259\) |
Rank
sage: E.rank()
The elliptic curves in class 1470.b have rank \(0\).
Complex multiplication
The elliptic curves in class 1470.b do not have complex multiplication.Modular form 1470.2.a.b
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.