Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-2811400x-1060760000\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-2811400xz^2-1060760000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3643575075x-49436164937250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2575, 92425)$ | $2.6252618861918286243271867475$ | $\infty$ |
| $(-400, 200)$ | $0$ | $2$ |
| $(1840, -920)$ | $0$ | $2$ |
Integral points
\( \left(-1184, 25288\right) \), \( \left(-1184, -24104\right) \), \( \left(-400, 200\right) \), \( \left(1840, -920\right) \), \( \left(2575, 92425\right) \), \( \left(2575, -95000\right) \), \( \left(11840, 1269080\right) \), \( \left(11840, -1280920\right) \)
Invariants
| Conductor: | $N$ | = | \( 124950 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $937140462562500000000$ | = | $2^{8} \cdot 3^{2} \cdot 5^{12} \cdot 7^{8} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{1336852858103281}{509796000000} \) | = | $2^{-8} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{-2} \cdot 17^{-2} \cdot 110161^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7234094539482327919135077970$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.94573542320352595206045175867$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9498836855302084$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.785511870448089$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.6252618861918286243271867475$ |
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| Real period: | $\Omega$ | ≈ | $0.12039821671018243188387426207$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2\cdot2\cdot2^{2}\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.5286147959576485665158966548 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.528614796 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.120398 \cdot 2.625262 \cdot 128}{4^2} \\ & \approx 2.528614796\end{aligned}$$
Modular invariants
Modular form 124950.2.a.g
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7077888 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $4$ | $I_{6}^{*}$ | additive | 1 | 2 | 12 | 6 |
| $7$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $17$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7140 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 3 & 2 \\ 2378 & 7139 \end{array}\right),\left(\begin{array}{rr} 3573 & 4 \\ 2 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5709 & 7138 \\ 5714 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1261 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1019 & 7138 \\ 0 & 7139 \end{array}\right),\left(\begin{array}{rr} 7137 & 4 \\ 7136 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[7140])$ is a degree-$7277201326080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7140\Z)$.
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 | $4$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 41650 = 2 \cdot 5^{2} \cdot 7^{2} \cdot 17 \) |
| $5$ | additive | $18$ | \( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \) |
| $7$ | additive | $32$ | \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 124950.g
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 3570.u2, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-35}, \sqrt{85})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-21}, \sqrt{-85})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{15}, \sqrt{21})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | nonsplit | add | add | ord | ord | split | ss | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 13 | 5 | - | - | 1 | 1 | 2 | 1,1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 1 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.