Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-82x+259\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-82xz^2+259z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6669x+208791\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(3, 7)$ | $0.62730543311029539265260323455$ | $\infty$ |
Integral points
\((3,\pm 7)\)
Invariants
| Conductor: | $N$ | = | \( 40432 \) | = | $2^{4} \cdot 7 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $283024$ | = | $2^{4} \cdot 7^{2} \cdot 19^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{10686208}{49} \) | = | $2^{8} \cdot 7^{-2} \cdot 13^{3} \cdot 19$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.10308152883986152827220403793$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.82487041888758337474611931706$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7882592414445331$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.342325682934626$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.62730543311029539265260323455$ |
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| Real period: | $\Omega$ | ≈ | $3.1007732427120086217653568474$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.8902638039925433302157143668 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.890263804 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 3.100773 \cdot 0.627305 \cdot 2}{1^2} \\ & \approx 3.890263804\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5184 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | -1 | 4 | 4 | 0 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $19$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
Galois representations
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 |
|---|---|---|
| $2$ | 2Cn | 2.2.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1596 = 2^{2} \cdot 3 \cdot 7 \cdot 19 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 1585 & 12 \\ 1584 & 13 \end{array}\right),\left(\begin{array}{rr} 1073 & 2 \\ 1122 & 13 \end{array}\right),\left(\begin{array}{rr} 913 & 12 \\ 690 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 1558 & 1581 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 936 & 5 \\ 895 & 1549 \end{array}\right),\left(\begin{array}{rr} 7 & 12 \\ 1530 & 1483 \end{array}\right),\left(\begin{array}{rr} 797 & 1584 \\ 0 & 1595 \end{array}\right)$.
The torsion field $K:=\Q(E[1596])$ is a degree-$11914076160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1596\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | additive | $2$ | \( 361 = 19^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 5776 = 2^{4} \cdot 19^{2} \) |
| $19$ | additive | $74$ | \( 112 = 2^{4} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 40432.m
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 10108.a2, its twist by $-4$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{19}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.3.361.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.41092625887488.3 | \(\Z/3\Z\) | not in database |
| $6$ | 6.6.158470336.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $18$ | 18.6.568840900549234150665808453878453293285376.2 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.69389168430559934753269628499830633398272.2 | \(\Z/2\Z \oplus \Z/6\Z\) | 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.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ord | ord | nonsplit | ss | ord | ord | add | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 11 | 1 | 1 | 1,1 | 1 | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 0 | 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.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.