Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-3107008x+2976964012\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-3107008xz^2+2976964012z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-251667675x+2169451761750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(14777, 1784250)$ | $7.2230528961245778125716427426$ | $\infty$ |
| $(-2123, 0)$ | $0$ | $2$ |
Integral points
\( \left(-2123, 0\right) \), \((14777,\pm 1784250)\)
Invariants
| Conductor: | $N$ | = | \( 29400 \) | = | $2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $-1906272111794400000000$ | = | $-1 \cdot 2^{11} \cdot 3^{10} \cdot 5^{8} \cdot 7^{9} $ |
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| j-invariant: | $j$ | = | \( -\frac{2568731006}{1476225} \) | = | $-1 \cdot 2 \cdot 3^{-10} \cdot 5^{-2} \cdot 1087^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7861805257933850431146228416$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.11335595772843332331390082726$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9759549111182807$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.554687357284239$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.2230528961245778125716427426$ |
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| Real period: | $\Omega$ | ≈ | $0.13726673857966439207557078147$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.9659396545576808156825163862 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.965939655 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.137267 \cdot 7.223053 \cdot 16}{2^2} \\ & \approx 3.965939655\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1290240 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | -1 | 3 | 11 | 0 |
| $3$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $7$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 837 & 4 \\ 836 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 737 & 106 \\ 104 & 735 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 337 & 4 \\ 674 & 9 \end{array}\right),\left(\begin{array}{rr} 281 & 4 \\ 562 & 9 \end{array}\right),\left(\begin{array}{rr} 124 & 1 \\ 599 & 0 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 419 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$5945425920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 175 = 5^{2} \cdot 7 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $18$ | \( 392 = 2^{3} \cdot 7^{2} \) |
| $7$ | additive | $20$ | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 29400.m
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 5880.b2, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-14}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.2469600.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.390331146240000.57 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.2775688151040000.25 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | add | nonsplit | add | add | ord | ord | ss | ord | ss | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 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 |
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.