Minimal Weierstrass equation
sage: E = EllipticCurve([0, -1, 1, -1746, -50295])
gp: E = ellinit([0, -1, 1, -1746, -50295])
magma: E := EllipticCurve([0, -1, 1, -1746, -50295]);
Minimal equation
Minimal equation
Simplified equation
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\(y^2+y=x^3-x^2-1746x-50295\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-x^2z-1746xz^2-50295z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2263248x-2373709104\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
sage: E.gens()
magma: Generators(E);
| $P$ | = |
\(\left(\frac{7045}{36}, \frac{573985}{216}\right)\)
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| $\hat{h}(P)$ | ≈ | $6.4244302828039167698966192616$ |
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: | \( 1859 \) | = | $11 \cdot 13^{2}$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $-777362416259 $ | = | $-1 \cdot 11^{5} \cdot 13^{6} $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{122023936}{161051} \) | = | $-1 \cdot 2^{12} \cdot 11^{-5} \cdot 31^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $0.97446483761236530333772945932\dots$ | ||
| Stable Faltings height: | $-0.30800984111840306468901426146\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
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| Analytic rank: | $1$ | ||
sage: E.regulator()
magma: Regulator(E);
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| Regulator: | $6.4244302828039167698966192616\dots$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | $0.35201532506737740953181959170\dots$ | ||
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 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | $1$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | $1$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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| Special value: | $ L'(E,1) $ ≈ $ 4.5229958287478482864024234053 $ | ||
Modular invariants
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);
\( q + 2 q^{2} - q^{3} + 2 q^{4} - q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{9} - 2 q^{10} - q^{11} - 2 q^{12} + 4 q^{14} + q^{15} - 4 q^{16} - 2 q^{17} - 4 q^{18} + O(q^{20}) \)
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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: | 2160 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
sage: E.local_data()
gp: ellglobalred(E)[5]
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $11$ | $1$ | $I_{5}$ | Non-split multiplicative | 1 | 1 | 5 | 5 |
| $13$ | $2$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
Galois representations
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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $5$ | 5Cs.4.1 | 5.60.0.1 |
$p$-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.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ss | ord | ord | ord | nonsplit | add | ord | ss | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2,3 | 1 | 1 | 1 | 1 | - | 1 | 1,1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0,0 | 0 | 1 | 0 | 0 | - | 0 | 0,0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 1859a
consists of 3 curves linked by isogenies of
degrees dividing 25.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/5\Z\) | 2.2.13.1-121.1-a2 |
| $3$ | 3.1.44.1 | \(\Z/2\Z\) | Not in database |
| $4$ | 4.0.21125.1 | \(\Z/5\Z\) | Not in database |
| $6$ | 6.0.21296.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
| $6$ | 6.2.4253392.1 | \(\Z/10\Z\) | Not in database |
| $8$ | 8.2.914519421387.3 | \(\Z/3\Z\) | Not in database |
| $8$ | 8.0.446265625.1 | \(\Z/5\Z \oplus \Z/5\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/4\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/10\Z\) | Not in database |
| $12$ | 12.0.2189052564185344.1 | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/15\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.