Minimal Weierstrass equation
sage: E = EllipticCurve([1, 1, 0, -25792533, -50429153907])
gp: E = ellinit([1, 1, 0, -25792533, -50429153907])
magma: E := EllipticCurve([1, 1, 0, -25792533, -50429153907]);
\(y^2+xy=x^3+x^2-25792533x-50429153907\)
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
sage: E.gens()
magma: Generators(E);
| \(P\) | = | \( \left(\frac{929133990936787885824953966304352174497}{4103960307326476516448996884296889}, \frac{28284711731918269289399085863042750131226072882735900709795}{262908560671210331216802572995372479438773107176787}\right) \) |
| \(\hat{h}(P)\) | ≈ | $88.641573920515742381528509674$ |
Integral points
sage: E.integral_points()
magma: IntegralPoints(E);
None
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 424830 \) | = | \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17^{2}\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-4429885005332640 \) | = | \(-1 \cdot 2^{5} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 17^{8} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{5551601340851881}{12960} \) | = | \(-1 \cdot 2^{-5} \cdot 3^{-4} \cdot 5^{-1} \cdot 7 \cdot 17 \cdot 35999^{3}\) |
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
BSD invariants
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sage: E.rank()
magma: Rank(E);
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| Analytic rank: | \(1\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(88.641573920515742381528509674\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.033496239879445527545322493765\) | ||
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sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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| Tamagawa product: | \( 2 \) = \( 1\cdot2\cdot1\cdot1\cdot1 \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(1\) | ||
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sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 424830.2.a.y
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
magma: ModularForm(E);
\( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + q^{10} + 4q^{11} - q^{12} + 5q^{13} + q^{15} + q^{16} - q^{18} - 3q^{19} + O(q^{20}) \)
For more coefficients, see the Downloads section to the right.
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sage: E.modular_degree()
magma: ModularDegree(E);
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| Modular degree: | 25704000 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
sage: r = E.rank();
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar = ellanalyticrank(E);
gp: ar[2]/factorial(ar[1])
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
\( L'(E,1) \) ≈ \( 5.9383188466663960980995509149112358626 \)
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
sage: E.local_data()
gp: ellglobalred(E)[5]
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(1\) | \(I_{5}\) | Non-split multiplicative | 1 | 1 | 5 | 5 |
| \(3\) | \(2\) | \(I_{4}\) | Non-split multiplicative | 1 | 1 | 4 | 4 |
| \(5\) | \(1\) | \(I_{1}\) | Non-split multiplicative | 1 | 1 | 1 | 1 |
| \(7\) | \(1\) | \(II\) | Additive | -1 | 2 | 2 | 0 |
| \(17\) | \(1\) | \(IV^{*}\) | Additive | -1 | 2 | 8 | 0 |
Galois representations
The 2-adic representation attached to this elliptic curve is surjective.
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) .
$p$-adic data
$p$-adic regulators
sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.
No Iwasawa invariant data is available for this curve.
Isogenies
This curve has no rational isogenies. Its isogeny class 424830y consists of this curve only.