Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-2819788x-1836647473\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-2819788xz^2-1836647473z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-45116603x-117590554858\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4307, 254557)$ | $3.4019160281002071965525503418$ | $\infty$ |
| $(1941, -971)$ | $0$ | $2$ |
Integral points
\( \left(1941, -971\right) \), \( \left(4307, 254557\right) \), \( \left(4307, -258865\right) \), \( \left(5525, 386101\right) \), \( \left(5525, -391627\right) \)
Invariants
| Conductor: | $N$ | = | \( 70070 \) | = | $2 \cdot 5 \cdot 7^{2} \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-23448206083143761920$ | = | $-1 \cdot 2^{20} \cdot 5 \cdot 7^{6} \cdot 11^{3} \cdot 13^{4} $ |
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| j-invariant: | $j$ | = | \( -\frac{21075830718885163521}{199306463150080} \) | = | $-1 \cdot 2^{-20} \cdot 3^{3} \cdot 5^{-1} \cdot 11^{-3} \cdot 13^{-4} \cdot 181^{3} \cdot 5087^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5380109698868058177332101087$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.5650558953591491651805337370$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0038593376627314$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.03585815232284$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.4019160281002071965525503418$ |
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| Real period: | $\Omega$ | ≈ | $0.058219772035436152169641401992$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 160 $ = $ ( 2^{2} \cdot 5 )\cdot1\cdot2^{2}\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.9223510255876188070459873932 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.922351026 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.058220 \cdot 3.401916 \cdot 160}{2^2} \\ & \approx 7.922351026\end{aligned}$$
Modular invariants
Modular form 70070.2.a.bq
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2580480 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $20$ | $I_{20}$ | split multiplicative | -1 | 1 | 20 | 20 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $13$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 40040 = 2^{3} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8324 & 34321 \\ 24983 & 11446 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 25033 & 7868 \\ 36470 & 39327 \end{array}\right),\left(\begin{array}{rr} 28599 & 0 \\ 0 & 40039 \end{array}\right),\left(\begin{array}{rr} 35043 & 17878 \\ 16450 & 19307 \end{array}\right),\left(\begin{array}{rr} 40033 & 8 \\ 40032 & 9 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 40034 & 40035 \end{array}\right),\left(\begin{array}{rr} 6868 & 34321 \\ 36631 & 11446 \end{array}\right),\left(\begin{array}{rr} 36961 & 34328 \\ 4844 & 17193 \end{array}\right)$.
The torsion field $K:=\Q(E[40040])$ is a degree-$10712468422656000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/40040\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 2695 = 5 \cdot 7^{2} \cdot 11 \) |
| $3$ | good | $2$ | \( 6370 = 2 \cdot 5 \cdot 7^{2} \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 7007 = 7^{2} \cdot 11 \cdot 13 \) |
| $7$ | additive | $26$ | \( 1430 = 2 \cdot 5 \cdot 11 \cdot 13 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 6370 = 2 \cdot 5 \cdot 7^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 70070bd
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1430h1, its twist by $-7$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-55}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{385}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{-55})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.17014071844000000.29 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | split | ss | nonsplit | add | nonsplit | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 5 | 3,1 | 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 | 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.