Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-2328818x-861559362\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-2328818xz^2-861559362z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3018148803x-40151641364802\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-689, 20753)$ | $2.0922103264089371800285035347$ | $\infty$ |
| $(-5157/4, 5157/8)$ | $0$ | $2$ |
Integral points
\( \left(-689, 20753\right) \), \( \left(-689, -20064\right) \), \( \left(3133, 148790\right) \), \( \left(3133, -151923\right) \)
Invariants
| Conductor: | $N$ | = | \( 75810 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $488381662908801810750$ | = | $2 \cdot 3 \cdot 5^{3} \cdot 7^{12} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{29689921233686449}{10380965400750} \) | = | $2^{-1} \cdot 3^{-1} \cdot 5^{-3} \cdot 7^{-12} \cdot 23^{3} \cdot 13463^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6712455026191449888304563166$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1990260130359247588259426007$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0394034516317685$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.948049991689934$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.0922103264089371800285035347$ |
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| Real period: | $\Omega$ | ≈ | $0.12571812522335185683102407982$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 1\cdot1\cdot1\cdot( 2^{2} \cdot 3 )\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.1563451177088235216054000765 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.156345118 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.125718 \cdot 2.092210 \cdot 48}{2^2} \\ & \approx 3.156345118\end{aligned}$$
Modular invariants
Modular form 75810.2.a.l
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3981312 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $7$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $19$ | $4$ | $I_0^{*}$ | 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 | 4.6.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 15960 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 19 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 10432 & 3363 \\ 5757 & 14194 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 3402 & 10279 \\ 9557 & 10052 \end{array}\right),\left(\begin{array}{rr} 13681 & 10944 \\ 2052 & 3649 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 14654 & 6731 \end{array}\right),\left(\begin{array}{rr} 11759 & 0 \\ 0 & 15959 \end{array}\right),\left(\begin{array}{rr} 15937 & 24 \\ 15936 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4846 & 7581 \\ 3705 & 10546 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 11496 & 13319 \\ 12065 & 9786 \end{array}\right)$.
The torsion field $K:=\Q(E[15960])$ is a degree-$22875026227200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/15960\Z)$.
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 | $4$ | \( 5415 = 3 \cdot 5 \cdot 19^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 722 = 2 \cdot 19^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 15162 = 2 \cdot 3 \cdot 7 \cdot 19^{2} \) |
| $7$ | split multiplicative | $8$ | \( 10830 = 2 \cdot 3 \cdot 5 \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 75810.l
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 210.d1, its twist by $-19$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{30}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-38}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-285}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{57}) \) | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{30}, \sqrt{-38})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{30}, \sqrt{57})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{-38})\) | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{57})\) | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.6480273456.6 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.432373800960000.13 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | 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 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.6.172424034338643537796234598856539207062500000000.1 | \(\Z/18\Z\) | 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 | nonsplit | nonsplit | split | ss | ord | ord | add | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 1 | 1 | 2 | 1,1 | 1 | 1 | - | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 2 | 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.