Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-1875x-17875\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-1875xz^2-17875z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2430675x-797519250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-19, 116\right) \) | $3.7599890408453402999193978255$ | $\infty$ |
| \( \left(-10, 5\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-19:116:1]\) | $3.7599890408453402999193978255$ | $\infty$ |
| \([-10:5:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-669, 23004\right) \) | $3.7599890408453402999193978255$ | $\infty$ |
| \( \left(-345, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-19, 116\right) \), \( \left(-19, -97\right) \), \( \left(-10, 5\right) \)
\([-19:116:1]\), \([-19:-97:1]\), \([-10:5:1]\)
\((-669,\pm 23004)\), \( \left(-345, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 1850 \) | = | $2 \cdot 5^{2} \cdot 37$ |
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| Minimal Discriminant: | $\Delta$ | = | $296000000000$ | = | $2^{12} \cdot 5^{9} \cdot 37 $ |
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| j-invariant: | $j$ | = | \( \frac{46694890801}{18944000} \) | = | $2^{-12} \cdot 5^{-3} \cdot 13^{3} \cdot 37^{-1} \cdot 277^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.89892056370294646308675815473$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.094201607485896275786378488117$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9139199638567892$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.54922196495469$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.7599890408453402999193978255$ |
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| Real period: | $\Omega$ | ≈ | $0.75158788142571371389547767916$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.8259621973928506635248894203 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.825962197 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.751588 \cdot 3.759989 \cdot 4}{2^2} \\ & \approx 2.825962197\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2304 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $5$ | $2$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $37$ | $1$ | $I_{1}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 4.6.0.3 | $6$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4440 = 2^{3} \cdot 3 \cdot 5 \cdot 37 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 2221 & 24 \\ 2220 & 1 \end{array}\right),\left(\begin{array}{rr} 1216 & 3 \\ 2301 & 4354 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 1111 & 24 \\ 12 & 289 \end{array}\right),\left(\begin{array}{rr} 3534 & 4439 \\ 3409 & 4432 \end{array}\right),\left(\begin{array}{rr} 4417 & 24 \\ 4416 & 25 \end{array}\right),\left(\begin{array}{rr} 7 & 24 \\ 492 & 1687 \end{array}\right),\left(\begin{array}{rr} 2961 & 4 \\ 2980 & 81 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 4 \\ 4340 & 4421 \end{array}\right)$.
The torsion field $K:=\Q(E[4440])$ is a degree-$167931740160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4440\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 925 = 5^{2} \cdot 37 \) |
| $3$ | good | $2$ | \( 925 = 5^{2} \cdot 37 \) |
| $5$ | additive | $18$ | \( 74 = 2 \cdot 37 \) |
| $37$ | nonsplit multiplicative | $38$ | \( 50 = 2 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 1850.f
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 370.d3, its twist by $5$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{185}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.2960.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{37})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.6325293375.3 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.299865760000.7 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.641431602250000.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.219040000.4 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.320074690238551125000000000000.3 | \(\Z/18\Z\) | 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 | nonsplit | ord | add | ord | ss | ord | ord | ord | ss | ord | ord | nonsplit | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | 1 | - | 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 | 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.