Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2+42850x+1835625\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z+42850xz^2+1835625z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+55532925x+84809922750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-165/4, 165/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 3675 \) | = | $3 \cdot 5^{2} \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $-6462652587890625$ | = | $-1 \cdot 3^{2} \cdot 5^{14} \cdot 7^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{4733169839}{3515625} \) | = | $3^{-2} \cdot 5^{-8} \cdot 23^{3} \cdot 73^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7219700489418357071180756019$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.055703981802871132734980436436$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0558519748642754$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.31225617961893$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.26981418112239574055064964830$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.0792567244895829622025985932 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.079256724 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.269814 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 1.079256724\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 18432 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $4$ | $I_{8}^{*}$ | additive | 1 | 2 | 14 | 8 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 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 |
|---|---|---|
| $2$ | 2B | 8.48.0.197 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 25 & 16 \\ 1624 & 2249 \end{array}\right),\left(\begin{array}{rr} 1343 & 952 \\ 0 & 3359 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1439 & 0 \\ 0 & 3359 \end{array}\right),\left(\begin{array}{rr} 1625 & 2912 \\ 182 & 2409 \end{array}\right),\left(\begin{array}{rr} 967 & 1456 \\ 3255 & 3193 \end{array}\right),\left(\begin{array}{rr} 3329 & 32 \\ 3328 & 33 \end{array}\right),\left(\begin{array}{rr} 1933 & 1456 \\ 1694 & 2745 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 4 & 129 \end{array}\right)$.
The torsion field $K:=\Q(E[3360])$ is a degree-$23781703680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3360\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $18$ | \( 147 = 3 \cdot 7^{2} \) |
| $7$ | additive | $26$ | \( 75 = 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 3675.j
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 15.a8, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{35}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-35}) \) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{35})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{6}, \sqrt{35})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{35})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.7965941760000.54 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.31116960000.31 | \(\Z/16\Z\) | not in database |
| $8$ | 8.2.6645780421875.3 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | 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 | \(\Z/24\Z\) | 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 |
|---|---|---|---|---|
| Reduction type | ord | nonsplit | add | add |
| $\lambda$-invariant(s) | 5 | 0 | - | - |
| $\mu$-invariant(s) | 1 | 0 | - | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.