Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-2450230x-1349948603\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-2450230xz^2-1349948603z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-39203675x-86435914250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-761, 8955)$ | $1.7697521313495202057940215040$ | $\infty$ |
| $(-1111, 555)$ | $0$ | $2$ |
Integral points
\( \left(-1111, 555\right) \), \( \left(-761, 8955\right) \), \( \left(-761, -8195\right) \), \( \left(2473, 86571\right) \), \( \left(2473, -89045\right) \), \( \left(3245, 156051\right) \), \( \left(3245, -159297\right) \)
Invariants
| Conductor: | $N$ | = | \( 31850 \) | = | $2 \cdot 5^{2} \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $153482061824000000000$ | = | $2^{20} \cdot 5^{9} \cdot 7^{8} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{884984855328729}{83492864000} \) | = | $2^{-20} \cdot 3^{3} \cdot 5^{-3} \cdot 7^{-2} \cdot 13^{-1} \cdot 32003^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6114911480246095863481207986$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.83381711727990274649506476027$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9692185652309827$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.3765835463938565$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.7697521313495202057940215040$ |
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| Real period: | $\Omega$ | ≈ | $0.12139975776910904716466072790$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 160 $ = $ ( 2^{2} \cdot 5 )\cdot2\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.5938992022878484203936474468 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.593899202 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.121400 \cdot 1.769752 \cdot 160}{2^2} \\ & \approx 8.593899202\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1105920 |
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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 4 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 |
| $5$ | $2$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $7$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $13$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 3119 & 0 \\ 0 & 3639 \end{array}\right),\left(\begin{array}{rr} 848 & 1043 \\ 3325 & 1562 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 3193 & 588 \\ 70 & 2927 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 2283 & 3318 \\ 1890 & 1107 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 3634 & 3635 \end{array}\right),\left(\begin{array}{rr} 412 & 2079 \\ 3409 & 3114 \end{array}\right),\left(\begin{array}{rr} 3633 & 8 \\ 3632 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[3640])$ is a degree-$811550638080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3640\Z)$.
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 | $4$ | \( 15925 = 5^{2} \cdot 7^{2} \cdot 13 \) |
| $5$ | additive | $18$ | \( 637 = 7^{2} \cdot 13 \) |
| $7$ | additive | $32$ | \( 650 = 2 \cdot 5^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 31850bs
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 910a1, its twist by $-35$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{65}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-455}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{65})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.46356673636000000.26 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.25969216000000.6 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.29668271127040000.144 | \(\Z/8\Z\) | 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 | add | add | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ss | ord |
| $\lambda$-invariant(s) | 5 | 1,1 | - | - | 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.