Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-1228483x-575017406\)
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(homogenize, simplify) |
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\(y^2z=x^3-1228483xz^2-575017406z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1228483x-575017406\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1577, 37544)$ | $1.0072379971455304178034490064$ | $\infty$ |
| $(1369, 17576)$ | $1.6553877326521635106388169645$ | $\infty$ |
Integral points
\((1369,\pm 17576)\), \((1577,\pm 37544)\), \((3735,\pm 216658)\), \((4697,\pm 311896)\), \((6270,\pm 488072)\), \((12887,\pm 1457326)\), \((67279,\pm 17448574)\)
Invariants
| Conductor: | $N$ | = | \( 75088 \) | = | $2^{4} \cdot 13 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $-24183269488299843584$ | = | $-1 \cdot 2^{13} \cdot 13^{7} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{1064019559329}{125497034} \) | = | $-1 \cdot 2^{-1} \cdot 3^{3} \cdot 13^{-7} \cdot 41^{3} \cdot 83^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4551611962465501046177529074$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.28979452610338456519600707000$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0626891964834324$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.797877354120879$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.6213691552529616488557034099$ |
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| Real period: | $\Omega$ | ≈ | $0.071229610687493235273444391117$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 56 $ = $ 2^{2}\cdot7\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $6.4674116477251811997812155383 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.467411648 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.071230 \cdot 1.621369 \cdot 56}{1^2} \\ & \approx 6.467411648\end{aligned}$$
Modular invariants
Modular form 75088.2.a.b
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2201472 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 4 | 13 | 1 |
| $13$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $19$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $7$ | 7B.6.3 | 7.24.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 13832 = 2^{3} \cdot 7 \cdot 13 \cdot 19 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 13819 & 14 \\ 13818 & 15 \end{array}\right),\left(\begin{array}{rr} 2183 & 0 \\ 0 & 13831 \end{array}\right),\left(\begin{array}{rr} 3457 & 10906 \\ 8911 & 7181 \end{array}\right),\left(\begin{array}{rr} 913 & 5624 \\ 3990 & 6385 \end{array}\right),\left(\begin{array}{rr} 6385 & 2926 \\ 4655 & 6651 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 6917 & 2926 \\ 8379 & 6651 \end{array}\right)$.
The torsion field $K:=\Q(E[13832])$ is a degree-$104081369333760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/13832\Z)$.
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 | $4$ | \( 4693 = 13 \cdot 19^{2} \) |
| $7$ | good | $2$ | \( 5776 = 2^{4} \cdot 19^{2} \) |
| $13$ | split multiplicative | $14$ | \( 5776 = 2^{4} \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 208 = 2^{4} \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 75088bj
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 26b2, its twist by $76$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.104.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1124864.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.7377869632.2 | \(\Z/7\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $14$ | 14.2.635693367356250458067828736.1 | \(\Z/7\Z\) | not in database |
| $18$ | 18.0.7939862701662327913375442615295910346752.1 | \(\Z/14\Z\) | 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 | ss | ord | ord | ord | split | ord | add | ord | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | - | 2,2 | 4 | 2 | 2 | 3 | 2 | - | 2 | 2 | 2 | 2 | 2,2 | 2 | 2 |
| $\mu$-invariant(s) | - | 0,0 | 0 | 1 | 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.